Super-hyperbolic orbits and non-collision singularities in a four-body problem

TL;DR

This paper proves the existence of super-hyperbolic orbits and noncollision singularities in the four-body problem, confirming longstanding conjectures and exploring their coexistence and implications for the classification of N-body motions.

Contribution

It establishes the existence of super-hyperbolic orbits and noncollision singularities in the four-body problem, solving two major conjectures and analyzing their coexistence.

Findings

Existence of super-hyperbolic orbits in four-body problem

Existence of noncollision singularities in four-body problem

Coexistence of both solution types for certain mass ratios

Abstract

In this paper, we prove the existence of super-hyperbolic orbits in four-body problem, which solves a conjecture of Marchal-Saari. We also prove the existence of noncollision singularities in the same model, which solves a conjecture of Anosov. Moreover, the two type of solutions coexist for certain mass ratios. We also make a conjecture on the classification of final motions of N-body problem, in which the superhyperbolic type of orbits is one of the building blocks.

Equations

$$\mathcal{H},\mathcal{P},\mathcal{B},\mathcal{O}\mathcal{S}$$

$$\mathcal{H},\mathcal{P},\mathcal{B},\mathcal{O}\mathcal{S},\mathcal{S}\mathcal{H}$$

$$\underset{t\to \infty }{\mathrm{limsup}}\underset{i,j}{\max }|Q_i(t)-Q_j(t)|=\infty ,\underset{t\to \infty }{\mathrm{liminf}}\underset{i,j}{\min }|Q_i(t)-Q_j(t)|=0.$$

$$\frac{1}{t}\max \{|Q_i(t)|,i=1,2,3,4\}\to \infty ,\mathrm{as}t\to \infty .$$

$$\sum _{i=1}^{4}d{P}_i\wedge d{Q}_i=\sum _{i=1}^{3}d{P}_i\wedge d{q}_i.$$

$$\{\begin{array}{ll}{\mathbf{x}}_0& =q_3\\ {\mathbf{x}}_1& =q_1-\frac{q_3}{2}\\ {\mathbf{x}}_2& =q_2-\frac{m_1{q}_1+q_3}{m_1+2}\end{array},\{\begin{array}{ll}{\mathbf{p}}_0& =P_3+\frac{P_1}{2}+\frac{P_2}{2}\\ {\mathbf{p}}_1& =P_1+\frac{m_1{P}_2}{m_1+2}\\ {\mathbf{p}}_2& =P_2\end{array}.$$

$$\sum _{i=3,1,2}d{P}_i\wedge d{q}_i=\sum _{i=0,1,2}d{\mathbf{p}}_i\wedge d{\mathbf{x}}_i.$$

$${\mathbf{x}}_0=(0,x_0{)}^t,{\mathbf{x}}_1=(x_1,0{)}^t,{\mathbf{p}}_0=(0,p_0{)}^t,{\mathbf{p}}_1=(p_1,0{)}^t,$$

$$M_0=\frac{1}{2},M_1=\frac{2m_1}{m_1+2}$$

$$\{\begin{array}{ll}r& =|\mathbf{x}|=\sqrt{M_0|{\mathbf{x}}_0{|}^2+M_1|{\mathbf{x}}_1{|}^2},\\ v& =r^{-1/2}(\mathbf{x}\cdot \mathbf{p}),\\ \psi & =\arctan \frac{\sqrt{M_1}{x}_1}{\sqrt{M_0}{x}_0},\\ w& =r^{-1/2}(\sqrt{M_0/M_1}{x}_0{p}_1-\sqrt{M_1/M_0}{x}_1{p}_0).\end{array}$$

$$\{\begin{array}{ll}{r}^{\prime }& =rv\\ v^{\prime }& =w^2+\frac{1}{2}{v}^2+\bar{V}(\psi ),\\ {\psi }^{\prime }& =w,\\ w^{\prime }& =-\frac{1}{2}vw-{\partial }_{\psi }\bar{V}(\psi ),\end{array}$$

$$\bar{V}(\psi )=-\frac{1}{\sqrt{2}\cos \psi }-\frac{4m_1}{\sqrt{2{\cos }^2\psi +\frac{2(2+m_1)}{m_1}{\sin }^2\psi }}.$$

$$rE=\frac{1}{2}{v}^2+\frac{1}{2}{w}^2+\bar{V}(\psi ),$$

$$(r,v,\psi ,w,L_2,{\ell }_2)\mapsto (\lambda r,v,\psi ,w,\sqrt{\lambda }{L}_2,{\ell }_2).$$

$$(L_0,{\ell }_0,L_1,{\ell }_1,L_2,{\ell }_2)\mapsto (\sqrt{\lambda }{L}_0,{\ell }_0,\sqrt{\lambda }{L}_1,{\ell }_1,\sqrt{\lambda }{L}_2,{\ell }_2);$$

$$(x_0,p_0;x_1,p_1;x_2,p_2)\mapsto (\lambda x_0,{\lambda }^{-1/2}{p}_0;\lambda x_1,{\lambda }^{-1/2}{p}_1;\lambda x_2,{\lambda }^{-1/2}{p}_2).$$

$$\{-E_0,E_1\}\subset (1-\mu ,1+\mu ),r_0,r_1ϵ,L_2/\sqrt{\lambda (\delta )}=O_{\delta \to 0}(1).$$

$$\sqrt{\mu }\le \frac{2m_2-m_1(m_1+m_2+2)}{(m_1+2)(m_1+m_2)}{\lambda }^{1/2}-\frac{8m_1{m}_2(m_1+m_2+2)}{(m_1+2{)}^2(m_1+m_2{)}^2}\le 10\sqrt{\mu }.$$

$$|{\mathbf{p}}_2|\in \frac{2m_2\sqrt{2M_1}}{m_1+m_2}(1-2\mu ,1+2\mu ),$$

$$d\mathcal{R}d\mathbb{L}=e^{-T{v}_{\ast }}{\mathbf{u}}_1\otimes {\mathbf{l}}_1+O(1),$$

$${\mathbf{u}}_1=(1,o(1),o(1),o(1)),{\mathbf{l}}_1=(o(1),1,o(1),O(1))$$

$$d\mathbb{G}=f_{\mu ,\lambda }\cdot {\chi }^2{\bar{\mathbf{u}}}_1\otimes {\bar{\mathbf{l}}}_1+O_{{\chi }^{-1}\to 0}(({ϵ}^2+{\lambda }^{-1/2})f_{\mu ,\lambda }\cdot \chi ),$$

$$(d_{\mathbf{x}}\mathbb{P}){\mathcal{C}}_{\eta }({\bar{\mathbf{u}}}_1)⊊{\mathcal{C}}_{\eta }({\bar{\mathbf{u}}}_1).$$

$$({\chi }_0\prod _{k=0}^{i-1}{\lambda }_k{)}^{-D}⩽{\delta }_i⩽{\delta }_{\ast },{\lambda }_0=1,{\lambda }_i=\lambda ({\delta }_i)={ϵ}^{-1}(2{\delta }_i{)}^{-1},i=1,2,\dots ,$$

$${\bar{t}}_i=O({\mu }^{-j(i)/2}{\mu }^{†}\prod _{k=1}^i({\lambda }_{\ast }/{\lambda }_k{)}^{1/2}{\chi }_0),$$

$$T_i=O(\sum _{\ell =1}^i{\mu }^{-j(\ell )/2}{\mu }^{†}\prod _{k=1}^{\ell }({\lambda }_{\ast }/{\lambda }_k{)}^{1/2}{\chi }_0).$$

$$\frac{\max \{|Q_j(t)|,j=1,2,3,4\}}{t}\sim {\lambda }_{\ast }^{i/2}={\lambda }_{\ast }^{O(\frac{1}{|\log \mu |}\log t)},\mathrm{as}t\to \infty .$$

$$H(P,Q)=\frac{P_1^2}{2m_1}+\frac{P_2^2}{2m_2}+\frac{P_3^2}{2}+\frac{P_4^2}{2}-\frac{m_1{m}_2}{|Q_1-Q_2|}-\sum _{i=1,2}^{j=3,4}\frac{m_i}{|Q_i-Q_j|}-\frac{1}{|Q_3-Q_4|}.$$

$$H_{:\cdot \cdot }(\mathbf{x},\mathbf{p})=\left(\frac{{\mathbf{p}}_1^2}{2M_1}+\frac{{\mathbf{p}}_0^2}{2M_0}-\frac{m_1}{|{\mathbf{x}}_1-\frac{{\mathbf{x}}_0}{2}|}-\frac{m_1}{|{\mathbf{x}}_1+\frac{{\mathbf{x}}_0}{2}|}-\frac{1}{|{\mathbf{x}}_0|}\right)$$

$$+\left(\frac{{\mathbf{p}}_2^2}{2M_2}-\frac{m_1{m}_2}{|{\mathbf{x}}_2-\frac{2{\mathbf{x}}_1}{m_1+2}|}-\frac{m_2}{|{\mathbf{x}}_2+\frac{m_1{\mathbf{x}}_1}{m_1+2}-\frac{{\mathbf{x}}_0}{2}|}-\frac{m_2}{|{\mathbf{x}}_2+\frac{m_1{\mathbf{x}}_1}{m_1+2}+\frac{{\mathbf{x}}_0}{2}|}\right)$$

Full text

1. 1 Introduction

2. 1.1 The configuration

3. 1.2 The statements of the results.

4. 1.3 Backgrounds and implications

5. 2 Preliminaries

6. 2.1 The Jacobi-Cartesian coordinates

7. 2.2 The isosceles three-body problem (I3BP)

8. 3 The isosceles four-body problem and proof of Theorem 1.3

9. 3.1 The local and global maps

10. 3.2 The key derivative estimate

11. 3.3 Proof of Theorem 1.3

12. 4 Estimate of the return time

13. 4.1 The Hamiltonian systems in Jacobi coordinates

14. 4.2 Transition from the left to the right

15. 4.3 Proof of Proposition 3.8

16. 5 The full four-body problem and proof of Theorem 1.4

17. 5.1 The coordinates and the diagonal form

18. 5.2 The I3BP near infinity

19. 5.3 The local map

20. 5.4 The global map

21. 5.5 The transversality condition

22. 5.6 Proof of Theorem 1.4

23. 6 Excluding double collisions

24. 6.1 Proof of Proposition 5.17

25. 6.2 Proof of the lemmas

26. A Delaunay coordinates

27. A.1 Elliptic motion

28. A.2 Hyperbolic motion

29. B The derivative of the global map in the I4BP

30. C The derivative of the global map in the F4BP

31. C.1 The $C^{1}$-dynamics of the variables $G_{i}$, $g_{i}$, $i=0,1,2$ in the isosceles limit

32. C.2 The non-isosceles case

33. D Numeric verification of the transversality conditions

1. Introduction

In this paper, we construct a model of four-body problem in which there exist both noncollision singularities and another kind of exotic solutions called superhyperbolic solutions, where for the latter the size of the system grows super-linearly with respect to time, i.e. $\frac{\max_{i}|Q_{i}(t)|}{t}\to\infty$ as $t\to\infty$. The study of noncollision singularities, i.e. finite time blowup solutions in $N$-body problem was motivated by the Painlevé conjecture. Two models of four-body problems with noncollision singularities were constructed in [X1, GHX] to which we refer readers for backgrounds on noncollision singularities. In this paper, we focus mainly on superhyperbolic solutions. This type of solutions was conjectured in [MS], which is defined globally in time, thus is not a singularity. However, since a superhyperbolic solution has velocity grows to infinity by definition, repeated accelerations are needed to construct such a solution. Therefore, we can consider superhyperbolic solutions as a slow-down version of noncollision singularities.

Our result in this paper solves the existence problem of superhyperbolic solutions. It is worthy of putting the result into a broader framework. Indeed, superhyperbolic solutions have an important position in the classification of final motions (asymptotic behaviors as $t\to\infty$) in $N$-body problem for $N\geq 4$. For three-body problem, it is well-known that there is a classification of the final motions given by Chazy (c.f. [AKN]) a hundred years ago, in which all logical possible combinations of the following building blocks

[TABLE]

is realized by an orbit, where $\mathcal{H}$ means hyperbolic, i.e. $|Q_{k}(t)|\to\infty,|\dot{Q}_{k}(t)|\to c_{k}>0$ for $k=1,2,3$, $\mathcal{P}$ means parabolic, i.e. $|Q_{k}(t)|\to\infty,\ |\dot{Q}_{k}(t)|\to 0$ for $k=1,2,3$, $\mathcal{B}$ means bounded, i.e. $\sup_{t}\max_{k}|Q_{k}(t)|<\infty$, $\mathcal{OS}$ means oscillatory, i.e. $\limsup|Q(t)|=\infty,\ \liminf|Q(t)|<\infty$, and $\mathcal{HP}$ means hyperbolic-parabolic i.e. some body is hyperbolic and some other body is parabolic asymptotically, similarly for $\mathcal{HB}$ and $\mathcal{PB}$ (c.f. [AKN, Chapter 2.3.4 ] for more details).

The classification of final motions in $N$-body problem for $N\geq 4$ is far less studied. We make the following conjecture, which claims that in addition to the above building blocks (1.1) for three-body problem, we only need to add one new block $\mathcal{SH}$, representing super-hyperbolic orbits.

Conjecture 1**.**

For $N$-body problem for $N\geq 4$, all logical possible combinations of

[TABLE]

is realized by an orbit and there is no other possibilities.

The conjecture is strongly supported by the following theorem of Marshal-Saari [MS].

Theorem 1.1**.**

For $N(>3)$-body problem as $t\to\infty$, there is a dichotomy: either it is a superhyperbolic orbit; or it decouples into several subsystems moving apart linearly and each subsystem grows at most like $O(t^{2/3})$.

This theorem leads to a possible approach to Conjecture 1 by analysing the growth rates. Indeed, letting $I(t)=\sum m_{i}|Q_{i}(t)|^{2}$ be the momentum of inertia of the system, Theorem 1.1 implies that $\sqrt{I}(t)/t=o(1)$ can only be at most $\sqrt{I}(t)=O(t^{2/3})$ as $t\to\infty$. Thus, Theorem 1.1 reduces Conjecture 1 to the following one.

Conjecture 2**.**

For $N$-body problem for $N\geq 4$, suppose a orbit satisfies $\sqrt{I}(t)=O(t^{2/3})$ as $t\to\infty$, then it is a combination of $\mathcal{P},\mathcal{B},\mathcal{OS}.$ Conversely, any logical possible combination of $\mathcal{P},\mathcal{B},\mathcal{OS}$ is realized by an orbit satisfying $\sqrt{I}(t)=O(t^{2/3})$ as $t\to\infty$.

As the complexity of possible final motions grows exponentially as $N$ gets large, we believe that a classification as asserted in Conjecture 1 gives a clean picture worthy of pursuing.

The following theorem is analogous to the classical von Zeipel theorem for noncollision singularities (see Appendix of [X2]).

Theorem 1.2**.**

Let $(Q_{i}(t),\dot{Q}_{i}(t))_{i=1}^{N}$ be a superhyperbolic solution of the $N$-body problem. Then we have as $t\to\infty$

[TABLE]

From this theorem, we can view a superhyperbolic orbit as a slow version of a noncollision singularity. The proof if in fact very simple. First, the $\limsup$ equation follows trivially from the definition of a superhyperbolic orbit. To prove the liminf one, we assume the liminf is $a>0$. Then in the Hamiltonian, the potential energy is bounded, thus by the energy conservation, the kinetic energy, hence the velocities are also bounded, so the orbit cannot be superhyperbolic, a contradiction.

1.1. The configuration

As we have discussed in the beginning of the paper, superhyperbolic orbits can be considered as a slow-down version of noncollision singularities. However, constructing such an orbit is by no means easy, since it requires more delicate control of the return times between two close encounters than noncollision singularities. As a result, it seems that superhyperbolic orbits are in some sense more delicate and rarer than noncollision singularities. In fact, we are not able to find superhyperbolic orbits in the models [X1, GHX]. Instead, we use here a new model and it is remarkable that in this model, superhyperbolic orbits and noncollision singularities coexist.

The configuration is as follows (see Figure 1): there is a particle $Q_{2}$ with mass $m_{2}$ moving to the right nearly along the $x$-axis and a binary $Q_{3}$-$Q_{4}$ with masses $m_{3}=m_{4}$ performing nearly Kepler elliptic motion when it is far away from other particles. The mass center of the binary moves to the opposite direction of $Q_{2}$ nearly along the $x$-axis. There is another particle $Q_{1}$ with mass $m_{1}$ traveling back and forth between the binary and $Q_{2}$ nearly along the $x$-axis. When $Q_{1}$ approaches the binary, a triple is formed and the triple will pass close to a triple collision after which all the three particles are significantly accelerated and the particle $Q_{1}$ is ejected to the right direction. It will quickly catch up with $Q_{2}$. Through two-body interaction, a large amount of the kinetic energy of $Q_{1}$ would transfer to $Q_{2}$. Then $Q_{1}$ would change the direction of movement and slowly get close to the binary. We need to carefully choose the mass ratios and the rate of acceleration to make sure this configuration would repeat itself infinitely many times and exists globally in time, so that the super-hyperbolic orbits could exhibit in this model.

1.2. The statements of the results.

We first consider the isosceles four-body problem by requiring $|Q_{i}-Q_{j}|=|Q_{7-i}-Q_{j}|$, $i=3,4,j=1,2$, where $Q_{1}$, $Q_{2}$ and the mass center of $Q_{3}$-$Q_{4}$ have always horizontal velocities. In this case, double collisions of the binary $Q_{3}$-$Q_{4}$ and of $Q_{1}$ and $Q_{2}$ are not avoidable. The collisions of these types can be regularized as elastic collisions. For this system, we have the following result.

Theorem 1.3**.**

There is an open set $\mathcal{A}\subset\mathbb{R}$ that contains 1 such that for each $m_{1}\in\mathcal{A}$ there exists $\lambda_{0}>1$ such that for each $\lambda\geqslant\lambda_{0}$, there exists an open set $\mathcal{B}_{\lambda}\subset\mathbb{R}$ such that if $m_{2}\in\mathcal{B}_{\lambda}$, then for the isosceles four-body problem with masses $(m_{1},m_{2})$ and $m_{3}=m_{4}=1$, we have

1. (1)

there exists a nonempty set $\Sigma_{sh}$ of initial conditions such that for all $x\in\Sigma_{sh}$, the orbit starting from $x$ exists globally in time upon the regularization of double collisions and

[TABLE] 2. (2)

there exists a non-empty set $\Sigma_{ncs}$ of initial conditions such that for all $x\in\Sigma_{ncs}$, there exists $t_{x}<+\infty$ such that as $t\to t_{x}$, $\max\{|Q_{i}(t)|,i=1,2,3,4\}\to\infty$.

Perturbing slightly away from the above solutions, we shall allow the binary to gain certain nonzero but very tiny angular of momentum. In this way, we avoid double collision. We obtain the following result on the existence of super-hyperbolic orbits and non-collision singularities in the model.

Theorem 1.4**.**

There is an open set $\mathcal{A}\subset\mathbb{R}$ that contains 1 such that for each $m_{1}\in\mathcal{A}$ there exists $\lambda_{0}>1$ such that for each $\lambda\geqslant\lambda_{0}$, there exists an open set $\mathcal{B}_{\lambda}\subset\mathbb{R}$ such that if $m_{2}\in\mathcal{B}_{\lambda}$, then for the full four-body problem with masses $(m_{1},m_{2})$ and $m_{3}=m_{4}=1$, we have

1. (1)

there exists a nonempty set $\Sigma_{sh}$ of initial conditions such that for all $x\in\Sigma_{sh}$, the orbit starting from $x$ exists globally in time without the occurrence of any collision between the particles and

[TABLE] 2. (2)

there exists a non-empty set $\Sigma_{ncs}$ of initial conditions such that for all $x\in\Sigma_{ncs}$, there exists $t_{x}<+\infty$ such that as $t\to t_{x}$, $\max\{|Q_{i}(t)|,i=1,2,3,4\}\to\infty$ and no collision happens before $t_{x}$.

In fact, for only the existence of non-collision singularities a much wider range of mass ratios is allowed, see Remark 3.14 and the discussion in Subsection 1.3 below.

In the last two theorems, by interpolating between noncollision singularities and superhyperbolic orbits, we can indeed get superhyperbolic orbits with $\frac{1}{t}\max\{|Q_{i}(t)|,i=1,2,3,4\}$ growing to infinity with arbitrarily high speed.

We discussed in [GHX] that the main theorem therein solves an analogue of a conjecture of Anosov claiming that a noncollision singularity can be found in a neighborhood of the singularities of [MM]. The model that we use in the present paper can be considered to be close to the model of [MM]. Thus, it is reasonable to consider statement (2) of Theorem 1.4 as a solution to the conjecture of Anosov.

In [GHX] we prove the existence of noncollision singularities in a similar model. There we consider a binary moving back and forth between two particles moving off to infinity in opposite directions. The mechanism of acceleration is similar to here, i.e. Devaney’s isosceles three-body problem. However, there are some important differences between the model of [GHX] and the model here:

1. (1)

The model in the present paper does not admit noncollision singularies with four equal masses in contrast to [GHX]. 2. (2)

During each return, the model of [GHX] accelerates twice. The rate of acceleration is so large that we do not know how to slow it down thus it is not clear to us whether it is possible to construct superhyperbolic orbit in the model of [GHX]. 3. (3)

A main difficulty in the proofs of the theorems in the present paper is to exclude double collision between $Q_{1}$ and $Q_{2}$, which is not present in the model of [GHX].

1.3. Backgrounds and implications

As we have discussed above, whereas near triple collision accelerates very intensively such that we can find noncollision singularities that accelerates arbitrarily fast, to slow down the orbit becomes a subtle problem.

The heuristic idea goes as follows. In Figure 1, the velocity of $Q_{1}$ after the triple collision can be accelerated arbitrarily and we fix such a ratio $\lambda$ throughout. When $Q_{1}$ comes close to $Q_{2}$, by momentum conservation, there is an energy transfer between the two bodies. Suppose the mass of $Q_{2}$ is much smaller than that of $Q_{1}$, then both will move towards the direction opposite to the binary so $Q_{1}$ will never return to interact with the binary for the next time. On the other hand, if the mass of $Q_{1}$ is much smaller than that of $Q_{2}$, then $Q_{1}$ will always return and a noncollision singularity may occur, which is the case for the model in [X1]. Thus, we expect that if we arrange the mass ratio of $Q_{2}$ and $Q_{1}$ carefully, we can control the return speed of $Q_{1}$ so small and its return time so long that to complete infinitely many returns, it takes infinite amount of time, which gives a superhyperbolic orbit.

This consideration leads to the following problem. It seems there is a critical mass ratio $\nu^{*}$ of $Q_{2}$ and $Q_{1}$ such that if $m_{2}/m_{1}>\nu^{*}$, then there is no superhyperbolic orbit.

**Problem: ** In this model, what is the critical ratio and what happens to the critical mass ratio, i.e. does superhyperbolic orbit exist or not?

Some readers may complain the conjectured picture of possible final motions for $N(>3)$-body problem in Conjecture 1 remains too complicated when the number of bodies grows. To pursue a simpler picture, it is reasonable to consider generic initial conditions or almost every initial data. Recall that it is conjectured that noncollision singularies has zero Lebesgue measure and first categroy. It is also natural to conjecture the same is true for superhyperbolic orbits. Finally, we recapitulate the following conjecture from [X2] and refer readers to find more discussions therein.

Conjecture 3**.**

For generic initial condition, the solution of the $N$-body problem is globally defined on $\mathbb{R}$, and as $t\to\pm\infty$, each body in the system approaches either a linear motion with constant velocity or a Kepler elliptic motion around a center moving linearly with constant velocity.

The paper is organized as follows. In Section 2, we introduce the coordinates: Jacobi coordinates and blowup coordinates, and the preliminaries on the isosceles three-body problem. In Section 3, we study the isosceles four-body probem and proof of Theorem 1.3. In Section 4, we give the Hamiltonians of the four-body problem and proves the key return time estimate. In Section 5, we prove the main theorem 1.4. Finally, we have four appendices. In Appendix A, we give some formulas for Delaunay coordinates. We give In Appendix B the proof of Proposition 3.10 on the derivative calculation for the $L,\ell$ parts and In Appendix C Proposition 5.10 on the derivative estimates for the $G$, $g$ parts . This calculation is very similar to that in [GHX], thus we mainly focus on the differences. At last, in Appendix D, we present a numeric verification of the non-degenerate conditions.

Acknowledgment

The authors are supported by grant NSFC (Significant project No.11790273) in China. J.X. is in addition supported by the Xiaomi endowed professorship of Tsinghua University.

2. Preliminaries

In this section we introduce several systems of coordinates for the four-body problem that we will use later. Without loss of generality, we assume $m_{3}=m_{4}=1$ throughout the paper and we always assume the binary $Q_{3}$-$Q_{4}$ is on the left, $Q_{2}$ is on the right and $Q_{1}$ travels between the binary and $Q_{2}$.

2.1. The Jacobi-Cartesian coordinates

The first step is to remove the translation invariance. We introduce $q_{i}=Q_{i}-Q_{4},\ i=3,1,2$. So we get that the following symplectic form is preserved by assuming $\sum_{i=1}^{4}P_{i}=0$

[TABLE]

We next assume that the particle $Q_{1}$ is closer to the binary than $Q_{2}$ and introduce is the following, which we call the left Jacobi coordinates,

[TABLE]

It can be easily checked that we have the following reduced symplectic form preserved by the coordinates change

[TABLE]

Clearly, $(\mathbf{x}_{0},\mathbf{p}_{0})$ describes the relative motion between the binary $Q_{3}$-$Q_{4}$, $(\mathbf{x}_{1},\mathbf{p}_{1})$ does that between $Q_{1}$ and the mass center of the binary, and $(\mathbf{x}_{2},\mathbf{p}_{2})$ does that between $Q_{2}$ and the mass center of the triple $Q_{1}$-$Q_{3}$-$Q_{4}$.

2.2. The isosceles three-body problem (I3BP)

In this section, we analyze the case when the triple $Q_{1}$-$Q_{3}$-$Q_{4}$ is close to triple collision and $Q_{2}$ is far apart. As a first approximation, we ignore $Q_{2}$ and focus only on the $Q_{1}$-$Q_{3}$-$Q_{4}$ three-body problem near triple collision. The problem is called the isosceles three-body problem first studied by Devaney [D]. In Jacobi coordinates we have

[TABLE]

We denote $\mathbf{x}=(M_{0}^{1/2}\mathbf{x}_{0},M_{1}^{1/2}\mathbf{x}_{1})\in\mathbb{R}^{4}$ and $\mathbf{p}=(M_{0}^{-1/2}\mathbf{p}_{0},M_{1}^{-1/2}\mathbf{p}_{1})\in\mathbb{R}^{4}$ and by $|\cdot|$ the Euclidean norm, and by

[TABLE]

the reduced masses. We next introduce the blowup coordinates

[TABLE]

The physical meanings of the variables are as follows. The variable $r$ measures the size of the I3BP, $v$ is the projection of the rescaled momentum $r^{1/2}\mathbf{p}$ to the radial componnet $\mathbf{x}$, $w$ is the scalar part of $\mathbf{w}$ where $\mathbf{w}=r^{1/2}\mathbf{p}-vr^{-1}\mathbf{x}$ is the projection of $r^{1/2}\mathbf{p}$ to the tangential of the sphere $|r^{-1}\mathbf{x}|=1$ and $\psi$ measures the relative size of the positions $x_{0}$ and $x_{1}$. All variables except $r$ are rescaling invariant and the triple collision corresponds to $r=0$. The coordinate change is accompanied by a time reparametrization $dt=r^{3/2}d\tau$. Equations of motion can be derived in terms of these coordinates

[TABLE]

where

[TABLE]

and the energy relation becomes

[TABLE]

where $E$ is the constant value of the Hamiltonian. In particular $r$-equation is $\frac{dr}{d\tau}=rv$, so we see that $\{r=0\}$ is an invariant submanifold. All the energy level sets are the same in the limit $r\to 0$, denoted by $\mathcal{M}_{0}$, has two dimensions. We call $\mathcal{M}_{0}$ the collision manifold and the dynamics on $\mathcal{M}_{0}$ is illustrated in Figure 2.

Theorem 2.1** ([D]).**

1. (1)

The collision manifold $\mathcal{M}_{0}$ is a topological 2-sphere with four punctures $($called arms$)$ with each puncture corresponding to $\psi=\pm\pi/2$. The manifold $\mathcal{M}_{0}$ is symmetric with respect to $\psi\mapsto-\psi$ or $w\mapsto-w$. 2. (2)

The variable $v$ is a Lyapunov function in the sense that $\frac{d}{d\tau}v\geq 0$ along the flow on $\mathcal{M}_{0}$ and $\frac{d}{d\tau}v=0$ iff at the fixed points. 3. (3)

*There are six fixed points on $\mathcal{M}_{0}$: the two with $(v,\psi,w)=(\pm v_{\dagger},0,0)$ correspond to Euler $($collinear$)$ central configurations and the four with $(v,\psi,w)=(\pm v_{*},\pm\psi_{*},0)$ correspond to Lagrange $($equilateral$)$ central configurations, where $v_{\dagger},v_{*}$ are numbers depending only on masses and $\psi_{*}=\arctan\left(\sqrt{\frac{3m_{1}}{2+m_{1}}}\right)$. *$($In the following, since we consider only the lower Lagrange point, we fix the convention v_{*}<0$$). 4. (4)

The Euler fixed point with $v>0$ is a sink and with $v<0$ is a source. For some masses, the eigenvalues are real and for others complex. The Lagrange fixed points are saddles.

We are only interested in the Lagrange fixed points in the lower half space $v<0$. The main observation is that, we can arrange the orbit to stay close to the saddle for as long time as we wish by selecting the initial condition sufficiently close to its stable manifold. Since in the lower half space we have $v_{*}<0$, from the equation $\frac{d}{d\tau}r=rv$ we get that $r$ will decrease to as small as we wish before leaving a neighborhood of triple collision. We shall consider the situation that the $Q_{1}$ comes from the right to have a near triple collision with the binary $Q_{3}$-$Q_{4}$. We hope that after the near triple collision the binary moves to the left and $Q_{1}$ moves to the right. In this case the relative position $x_{1}$ from the mass center of the binary to $Q_{1}$ has a positive sign before and after the near triple collision, therefore the variable $\psi$ should also carry a positive sign before and after the near triple collision. Thus we need the right lower Lagrange fixed point to have a stable manifold coming from the right lower arm and an unstable manifold escaping to the right upper arm. The existence of the stable manifold coming from the right lower arm follows directly from the fact that $v$ is a Lyapunov function. However, the existence of an unstable manifold escaping to the right upper arm is nontrivial, which depends on the mass ratio $m_{1}/m_{3}=m_{1}$. This was studied by [SM]. Here we only cite the relevant statements.

Theorem 2.2** ([SM]).**

Assume $m_{1}<55/4$ in which case the Euler fixed points are respectively sink and source with complex eigenvalues. Then there exist $b_{1}\simeq 0.379$ and $b_{2}\simeq 2.662$ such that the following holds for the right lower Lagrange fixed point:

1. (1)

when $b_{1}<m_{1}<b_{2}$, the two unstable manifolds escape from the two upper arms respectively; 2. (2)

when $m_{1}>b_{2}$, one of the unstable manifold escapes from the right upper arm and the other dies at the upper Euler fixed point.

Notation 2.3**.**

1. (1)

Denote by $O$ the right lower Lagrange fixed point on the collision manifold of $Q_{1}$-$Q_{3}$-$Q_{4}$. 2. (2)

Denote by $W^{s}_{\therefore}(O)$ the stable manifold of $O$, in the isosceles three-body problem $Q_{1}$-$Q_{3}$-$Q_{4}$. We see that $W^{s}_{\therefore}(O)$ has have dimension two. 3. (3)

Denote by $W^{u}_{0}(O)$ the branch of the unstable manifold of $O$ on the collision manifold $\mathcal{M}_{0}$ that escapes from the right upper arm.

Note the fact the an orbit starting sufficiently close to the stable manifold $W^{s}_{\therefore}(O)$ of the Lagrange fixed point $O$ will follow the unstable manifold of that point very far. There are two branches of the unstable manifold on $\mathcal{M}_{0}$ and there are two sides of the two dimensional manifold $W^{s}_{\therefore}(O)$ in the three dimensional energy level set. In order for the exiting orbit to follow the correct branch of the unstable manifold $W^{u}_{0}(O)$ defined above, we have to choose initial condition on the correct side of $W^{s}_{\therefore}(O)$. In the rest of the paper we use the phrase “correct side” of the stable manifold $W^{s}_{\therefore}(O)$ for this meaning.

3. The isosceles four-body problem and proof of Theorem 1.3

In this section we consider the isosceles four-body problem (I4BP) and prove Theorem 1.3. We first introduce the Poincaré sections $\mathcal{S}_{\pm}$ to separate the near triple collision regime and the perturbed Kepler motions regime.

Definition 3.1**.**

Let us fix $\epsilon>0$ to be a sufficiently small number whose meaning will be reserved throughout the paper. We introduce a Poincaré section $\mathcal{S}_{-}:=\{r=\epsilon^{-1}\}$ before the near triple collision and $\mathcal{S}_{+}=\{v=\epsilon^{-1/2}\}$ after the near triple collision.

3.1. The local and global maps

Using the sections $\mathcal{S}_{\pm}$, we introduce the return maps $\mathbb{L}$ and $\mathbb{G}$ called local map and global map respectively.

Definition 3.2** (Local and global maps).**

We define the local map $\mathbb{L}$ to be the Poincaré map going from $\mathcal{S}_{-}$ to $\mathcal{S}_{+}$, and global map $\mathbb{G}$ to be the Poincaré map going from $\mathcal{S}_{+}$ to $\mathcal{S}_{-}$.

For the global map piece of the orbit, we treat $(\mathbf{x}_{0},\mathbf{p}_{0})$ as a degenerate Kepler elliptic motion and introduce Delaunay coordinates $(L_{0},\ell_{0})$ for it, and treat $(\mathbf{x}_{1},\mathbf{p}_{1})$ and $(\mathbf{x}_{2},\mathbf{p}_{2})$ as degenerate Kepler hyperbolic motions and introduce Delaunay coordinates $(L_{1},\ell_{1})$ and $(L_{2},\ell_{2})$ respectively for them. The isosceles assumption guarantees that $G_{0}=g_{0}-\frac{\pi}{2}=G_{1}=g_{1}=G_{2}=g_{2}=0$. For the local map piece of orbit, we convert $(\mathbf{x}_{0},\mathbf{p}_{0},\mathbf{x}_{1},\mathbf{p}_{1})$ into blowup coordinates $(r,v,\psi,w)$ and $(\mathbf{x}_{2},\mathbf{p}_{2})$ into Delaunay coordinates $(L_{2},\ell_{2})$.

Notation 3.3**.**

We use the super-script $i$ and $f$ to stand for the corresponding variables on the initial and final sections respectively.

The following lemma says that the local map is well approximated by the dynamics of the isosceles three-body problem $Q_{1}$-$Q_{3}$-$Q_{4}$.

Proposition 3.4**.**

For the I4BP, there exist $C_{1}>1,D>0,\chi_{0}:=\chi_{0}(m_{1})\gg 1$ and $\delta_{0}:=\delta_{0}(m_{1})>0$ such that for any $\chi>\chi_{0}$, $\delta\in[\chi^{-D},\delta_{0}]$ and $\mu\in(0,1)$ the following holds: Let $\zeta:\ [0,1]\to\mathcal{S}_{-}$ be a smooth curve on the section $\mathcal{S}_{-}$ satisfying

1. (1)

$\zeta$* is on the correct side of $W^{s}_{\therefore}(O)$;* 2. (2)

$\zeta(0)\in W^{s}_{\therefore}(O)$* and $\mathrm{dist}(\zeta(1),W^{s}_{\therefore}(O))=\epsilon^{2}$;* 3. (3)

on the curve $\zeta$, $|\mathbf{x}_{2}|\geq\chi$ and $|\mathbf{p}_{2}|=O(1)$.

Then

1. (1)

there exists a subsegment $\bar{\zeta}$ of $\zeta$, such that for any point $\mathbf{x}$ on the image of the curve $\bar{\zeta}$ under the local map, its $r$-coordinate satisfies $r^{f}\in[(1-\mu)\delta,(1+\mu)\delta]$. 2. (2)

the oscillation of $L_{2}$ is estimated as $|L_{2}^{f}-L_{2}^{i}|\leq\frac{C_{1}}{\chi^{3}}$, and 3. (3)

the travel time for points on $\bar{\zeta}$ between the two sections satisfies $C_{1}^{-1}\log\delta^{-1}<|\tau^{f}-\tau^{i}|<C_{1}\log\delta^{-1}$, and in the original time scale $t$, we have $|t^{f}-t^{i}|\leqslant C_{1}$.

We refer the reader to [GHX, Section 6.1] for a proof of this statement. Here we only outline the main idea. For the I3BP, if a curve $\zeta$ is transverse to the stable manifold $W^{s}_{\therefore}(O)$, the closer of a point on $\zeta$ to $W^{s}_{\therefore}(O)$, the longer its orbit stays in a neighborhood of $O$. From the $r^{\prime}=rv$ equation, we get that the longer the orbit stays close to $O$ where $v=v_{*}<0$, the more $r$ decreases. So it is possible to select a subsegment on $\zeta$ such that $r^{f}$ lies in the given window $[(1-\mu)\delta,(1+\mu)\delta]$. Since $Q_{2}$ is far, its perturbation to the I3BP can be estimated to be small.

Immediately after the local map, on the sections $\mathcal{S}_{+}$, we have that $r^{f}$ is small and the velocities are large. So we introduce the renormalization map to zoom in the spatial variables and to slow down the velocities as follows.

Definition 3.5** (Renormalization map).**

Let $\lambda:=\lambda(\delta)=\frac{\epsilon^{-1}}{2\delta}$, where $\delta\leqslant\delta_{0}$ is as in Proposition 3.4. The renormalization map $\mathcal{R}_{\lambda}$ on the section $\mathcal{S}_{+}$ is as follows,

[TABLE]

The effect of the renormalization on the Delaunay coordinates and the Cartesian coordinates are as follows respectively

[TABLE]

Moreover, we also make the time change $t\mapsto t\lambda^{3/2}$ and the change of Hamiltonian $H\mapsto H/\lambda$ when applying the renormalization.

The main effect of the renormalization is to rescale the semimajor of the binary’s elliptic motion to order 1 size. Indeed, since $r$ is of order $\delta$ after the local map, the renormalization stretches $r$ to order $1/\epsilon$. Since on $\mathcal{S}_{+}$, we have $v=\epsilon^{-1/2}$. Using the definition of $v$, we see that $\mathbf{p}$ is of order 1, thus both subsystems $(\mathbf{x}_{0},\mathbf{p}_{0})$ and $(\mathbf{x}_{1},\mathbf{p}_{1})$ have order 1 energies. Applying the renormalization to the outcome of local map in Proposition 3.4, we have the following.

Proposition 3.6**.**

After applying the renormalization on the section $\mathcal{S}_{+}$, for the image $\mathcal{R}_{\lambda(\delta)}\mathbb{L}\bar{\zeta}$ where $\bar{\zeta}$ is the resulting segment in Proposition 3.4, we have

[TABLE]

Here $E_{0}:=-\frac{M_{0}}{2L_{0}^{2}}$, $E_{1}:=\frac{2m^{2}_{1}M_{1}}{L^{2}_{1}}$ are the energy for $(\mathbf{x}_{i},\mathbf{p}_{i})$, $i=0,1$ as defined in (4.3) and $M_{i}$, $i=0,1$ are the reduced masses as in (2.2).

Readers can find a proof of this statement in [GHX, Section 6.2].

After the local map and the renormalization, $Q_{1}$ would move to the right with speed of order 1, and $Q_{2}$ moves with speed of order $\lambda^{-1/2}$. When $Q_{1}$ catches up with $Q_{2}$, we want that after the two-body interaction between $Q_{1}$ and $Q_{2}$, a large part of the kinetic energy of $Q_{1}$ would transfer to $Q_{2}$, such that $Q_{2}$ continues moving to the right with speed of order 1 and $Q_{1}$ is bounced back to the left with much smaller velocity: the relative speed between $Q_{1}$ and the mass center of the binary $Q_{3}$-$Q_{4}$ is of order $\lambda^{-1/2}$. This could be achieved by carefully choosing the mass ratio of $Q_{1}$-$Q_{2}$ and the initial speed of $Q_{2}$ as follows.

Definition 3.7**.**

Let $m_{1}\in(b_{1},2)$, $0<\mu\ll 1$ and $\lambda\gg 1$. We say that $m_{2}$ is $(m_{1},\mu,\lambda)$-admissible if we have

[TABLE]

In the limit $\mu\to 0$ and $\lambda\to\infty$, we have $m_{2}\to\frac{m_{1}^{2}+2m_{1}}{2-m_{1}}$. Thus solution exists for $m_{1}\in(b_{1},2)$ and $\mu,\lambda^{-1}$ small enough and the admissible $m_{2}$’s form an interval of length $O(\sqrt{\mu\lambda^{-1}})$. Note that for $m_{1}\geqslant 2$ the above condition will never be satisfied. This is natural, since if $m_{1}\geqslant 2$, then when ejected from near triple collision, the absolute velocity of $Q_{1}$ would be smaller than that of the mass center of the binary. In such scenario, $Q_{1}$ would never catch up with the binary after being reflected by $Q_{2}$.

If the mass ratio is chosen as in the last definition, then we have the following estimate of the distance and travel time for $Q_{1}$ to reach the next $\mathcal{S}_{-}$ section.

Proposition 3.8**.**

Let $m_{1}\in(b_{1},2)$ $0<\mu\ll 1$, and $\lambda_{*},m_{2}$ be such that $m_{2}$ is $(m_{1},\mu,\lambda_{*})$-admissible $($respectively, in addition $\lambda\geq 2\lambda_{*})$. We choose $\epsilon$ in the definition of the sections $\mathcal{S}_{\pm}$ to be smaller than $\mu$. Suppose on the section $\mathcal{S}_{+}$ before renormalization $\mathcal{R}_{\lambda(\delta)}$ we have

[TABLE]

and (3.2) are satisfied after renormalization $\mathcal{R}_{\lambda_{*}}$ $($respectively $\mathcal{R}_{\lambda})$. Then

1. (1)

the particle $Q_{1}$ ejected from near triple collision as in Proposition 3.6 would come to a two-body interaction with $Q_{2}$ after which $Q_{1}$ would return to the section $\mathcal{S}_{-}$. 2. (2)

*After renormalization $\mathcal{R}_{\lambda_{*}}$ *$($*respectively $\mathcal{R}_{\lambda})$, the timespan of these orbits from section $\mathcal{S}_{+}$ to $\mathcal{S}_{-}$ is $O_{\chi^{-1}\to 0}(\mu^{-1/2}\lambda_{*}^{1/2}\chi)$ *$($*respectively $O_{\chi^{-1}\to 0}(\lambda_{*}^{1/2}\chi))$ and when reaching $\mathcal{S}_{-}$, (3.4) is satisfied and the distance between $Q_{1}$ and $Q_{2}$ is $O_{\chi^{-1}\to 0}(\mu^{-1/2}\lambda_{*}^{1/2}\chi)$ *$($*respectively $O_{\chi^{-1}\to 0}(\lambda_{*}^{1/2}\chi))$, where $\chi$ is the initial distance between $Q_{1}$ and $Q_{2}$ on the section $\mathcal{S}_{+}$ after renormalization $\mathcal{R}_{\lambda_{*}}$ *$($respectively $\mathcal{R}_{\lambda})$.

We will prove this proposition in Section 4.3.

3.2. The key derivative estimate

On the sections $\mathcal{S}_{\pm}$, we perform the standard energetic reduction to eliminate $(L_{1},\ell_{1})$ from Delaunay coordinates by fixing the total energy and treating the variable $\ell_{1}$ as the new time. So we use the four variables $(L_{0},\ell_{0},L_{2},\ell_{2})$ to parametrize the sections $\mathcal{S}_{\pm}$. We have the following estimates of the derivatives of the local and global maps. In both cases, we use Delaunay coordinates. Denote by $\mathcal{M}$ the phase space of the isosceles four-body problem (I4BP), which has six dimensions.

Proposition 3.9** (Proposition 3.11 of [GHX]).**

Let $\gamma:\ [0,T]\to\mathcal{M}$ be an orbit of the I4BP with $\gamma(0)\in\mathcal{S}_{-}$ and $\gamma(T)\in\mathcal{S}_{+}$ as in Proposition 3.4. Then for $\delta$ sufficiently small, we have $T=O(\log\delta^{-1})$ and the following estimate of the derivatives of the local map along $\gamma$

[TABLE]

where using the variables $(L_{0},\ell_{0},L_{2},\ell_{2}),$ we have

[TABLE]

as $\delta\to 0$ and $\chi\to\infty$.

We refer readers to [GHX] for a proof. The proposition implies that the only dominant term in $d\mathcal{R}d\mathbb{L}$ is given by $\frac{\partial L^{f}_{0}}{\partial\ell_{0}^{i}}$. Since we can parametrize the curve $\zeta$ in Proposition 3.4 by $\ell_{0}$ as a substitute of $\psi$, and $L_{0}^{2}$ has the meaning of the semimajor of the elliptic motion, the proposition means that by changing the initial point on $\zeta$ slightly, we can get a significant change of the semimajor of the binary on the section $\mathcal{S}_{+}$, which follows from a similar analysis to Proposition 3.4 of the hyperbolic dynamics near the Lagrange fixed point.

We next estimate the derivative of the global map.

Proposition 3.10**.**

There exists $\chi_{0}\gg 1$ such that the following holds: Let $\gamma:\ [0,T]\to\mathcal{M}$ be an orbit of the I4BP as in Proposition 3.8 with $\gamma(0)\in\mathcal{S}_{+}$ and $\gamma(T)\in\mathcal{S}_{-}$ and the initial condition $\gamma(0)$ satisfying (3.2) and in addition $|\mathbf{x}_{2}|=\chi\geq\chi_{0}.$ Then we have the following derivative estimate of the derivatives of the global map along $\gamma$

[TABLE]

where $f_{\mu,\lambda}=\mu^{-1/2}\lambda_{*}^{1/2}$ if $\lambda=\lambda_{*}$ and $=\lambda_{*}^{1/2}$ if $\lambda>2\lambda_{*}$ corresponding to Proposition 3.8, $\bar{\mathbf{u}}_{1}=(0,1,0,O(1))$ and $\bar{\mathbf{l}}_{1}=(1,0,0,0)$.

The proof of this proposition will be given in Appendix B. The main dominant entry that will play an important role in the proof is $\frac{\partial\ell_{0}^{f}}{\partial L_{0}^{i}}$, which means that a small change in the semimajor of the binary on $\mathcal{S}_{+}$ will be stretched to a huge phase difference $\ell_{0}$ on the section $\mathcal{S}_{-}$ after the global map.

From the last two propositions, it is clear that we have the transversality condition $\mathbf{l}_{1}\cdot\bar{\mathbf{u}}_{1}\neq 0$ and $\bar{\mathbf{l}}_{1}\cdot\mathbf{u}_{1}\neq 0$. We define $\mathbb{P}:=\mathbb{G}\mathcal{R}\mathbb{L}:\ \mathcal{S}_{-}\to\mathcal{S}_{-}$ the Poincaré map. Thus we have the following cone preservation property.

Notation 3.11** (Cone).**

1. (1)

Let $\mathbf{u}_{1},\ldots,\mathbf{u}_{n}$ be a tuple of linearly independent vectors, we denote by $\mathcal{C}_{\eta}(\mathbf{u}_{1},\ldots,\mathbf{u}_{n})$ the $\eta$-cone around $\mathbf{u}_{1},\ldots,\mathbf{u}_{n}$ that is the set of vectors forming an angle at most $\eta$ with the plane span$\{\mathbf{u}_{1},\ldots,\mathbf{u}_{n}\}$.

We fix a small number $\eta>0$ and choose $\delta$ small and $\chi$ large accordingly.

Proposition 3.12**.**

Let $\mathbf{x}\in\mathcal{S}_{-}$ be the initial condition for an orbit of the I4BP such that $\mathbb{P}(\mathbf{x})\in\mathcal{S}_{-}$. Suppose the assumptions of Proposition 3.9 and 3.10 are satisfied along the orbit, then we have

[TABLE]

Moreover, for each $v\in\mathcal{C}_{\eta}(\bar{\mathbf{u}}_{1})$, we have $|d_{\mathbf{x}}\mathbb{P}(v)|\geq f_{\mu,\lambda}\cdot e^{-Tv_{*}}\chi^{2}|v|$.

3.3. Proof of Theorem 1.3

We now give the proof of Theorem 1.3 assuming the propositions in the previous subsections. We first show that there exists a Cantor set of initial conditions such that the map $\mathbb{P}$ can be iterated for infinitely many steps with prefixed renormalization maps $\mathcal{R}_{\lambda(\delta_{i})}$, $i\in\mathbb{N}$. Then we show that such initial conditions indeed lead to super-hyperbolic orbits or non-collision singularities (with double collisions regularized) depending on the choice of $\{\delta_{i}\}$.

Proof of Theorem 1.3.

From now on we only consider the phase points on $\mathcal{S}_{+}$ such that the conditions in Proposition 3.8 and (3.2) are satisfied. Clearly, they form an open set on the section $\mathcal{S}_{+}$. Fix a small number $\eta>0$, for any such a phase point $X=(L_{0},\ell_{0},L_{2},\ell_{2})\in\mathcal{S}_{+}$, we introduce the cone $\mathcal{C}_{\eta}(\mathbf{u}_{1})\subset T_{X}\mathcal{S}_{+}$.

We have the following non-degeneracy property for the global map.

Lemma 3.13**.**

Let $\xi$ be an initial segment of length $\frac{1}{\chi}$ on the section $\mathcal{S}_{+}$ with all its tangent vectors lying inside the cone $\mathcal{C}_{\eta}(\bar{\mathbf{u}}_{1})$ for each point $X$ on $\xi$. Then its image under the global map on the section $\mathcal{S}_{-}$ wind around the cylinder formed by $(L_{0},\ell_{0})$ along the direction $\ell_{0}$ many times. In particular, it intersects transversely with the stable manifold $W_{\therefore}^{s}(O)$ when projected to $\mathcal{M}_{\therefore}$.

We refer the reader to [GHX, Section 5.5] for an argument working exactly the same for proving this statement.

Step 1: Construction of the Cantor set.

Recall that we fix $m_{1}\in(b_{1},2)$, $0<\mu\ll 1$, $\delta_{*}(\leq\delta_{0})$ sufficiently small, $\lambda_{*}=\epsilon^{-1}(2\delta_{*})^{-1}$ and $m_{2}$ to be $(m_{1},\mu,\lambda_{*})$-admissible. Let us choose a sequence $\{\delta_{i}\}$ such that

[TABLE]

where $\chi_{0}$ and $D$ are as in Proposition 3.4, and we assume either $\delta_{i}=\delta_{*}$ or $\delta_{i}\leq\delta_{*}/2$. Then the assumptions in Proposition 3.4 is satisfied for $\delta=\delta_{i}$ and $\chi_{i-1}=\chi_{0}\prod_{k=0}^{i-1}\lambda_{k}$ after the application of the the $i$-th renormalization map $\mathcal{R}_{\lambda_{i}}$, $i=1,2,3,\dots$.

We start with an initial segment $\zeta_{0}$ on the section $\mathcal{S}_{-}$ from Proposition 3.4 with the chosen $\delta=\delta_{1}$.

We know that for the image $\mathbb{L}\zeta_{0}$ of the curve $\zeta_{0}$ under the local map to the section $\mathcal{S}_{+}$, we can select a subsegment $\bar{\zeta}_{0}\subset\zeta_{0}$ such that each point in the image $\mathbb{L}\bar{\zeta}_{0}$ has $r$-component ranging from $(1-\mu)\delta_{1}$ to $(1+\mu)\delta_{1}$.

Applying the renormalization map to $\mathbb{L}\bar{\zeta}_{0}$. Then the segment is rescaled to length of order one and the $r$-component has values in the interval $(\frac{1}{2}(1-\mu)\epsilon^{-1},\frac{1}{2}(1+\mu)\epsilon^{-1})$ and the phase points on it satisfy the condition (3.2). The renormalization map stretches the $L_{0}$ and $L_{2}$ variables with the same ratio $\lambda_{1}^{1/2}$ and leaves $\ell_{0}$, $\ell_{2}$ untouched. Therefore after the renormalization, the segment $\mathcal{R}\mathbb{L}\bar{\zeta}_{0}$ has tangent vectors lying in the cone $\mathcal{C}_{\eta}(\mathbf{u}_{1})$ by Proposition 3.9, provided the rescaling factor $\lambda_{1}\gg\eta^{-1}$.

Next, we apply the global map to obtain the strong expansion in Proposition 3.12. The strong expansion shows that after applying the global map $\mathbb{G}$ and arriving at the section $\mathcal{S}_{-}$, the resulting curve $\mathbb{P}\bar{\zeta}_{0}$ winds around the $\ell_{0}$-circle for many times. Then on the section $\mathcal{S}_{-}$, we are in a position to pick from the image $\mathbb{P}\bar{\zeta}_{0}$ segments with phase points satisfying the assumptions of Proposition 3.4 with $\delta=\delta_{2}$ as in (3.7). This involves deleting many open intervals from the segment $\mathcal{R}\mathbb{L}\bar{\zeta}_{0}$, and correspondingly in the original segment $\bar{\zeta}_{0}$. We then repeat the above procedure to the newly picked segments on the section $\mathcal{S}_{-}$. Repeating this procedure for infinitely many steps, in the limit, we get a Cantor set $\mathcal{I}_{\zeta_{0},\{\delta_{i}\}}$ on the curve $\zeta_{0}$ as a result of deleting open intervals for infinitely many steps.

Step 2: Time estimate for super-hyperbolic orbit or non-collision singularity.

We next show that each initial condition in the Cantor set $\mathcal{I}_{\zeta_{0},\{\delta_{i}\}}$ leads to a super-hyperbolic orbit or a non-collision singularity depending on the choice of $\{\delta_{i}\}$.

Consider $\mathsf{x}_{0}\in\mathcal{I}_{\zeta_{0},\{\delta_{i}\}}$ and assume that the initial distance between the particles $Q_{2}$ and the mass center of the binary $Q_{3}$-$Q_{4}$ is $\chi_{0}$ as in Proposition 3.4. Suppose $Q_{1}$ comes to a near triple collision with the binary. Then after the local map and the first renormalization $\mathcal{R}_{\lambda_{1}}$, we have the distance between $Q_{1}$ and $Q_{2}$ is $\chi_{1}=O(\lambda_{1}\chi_{0})$ and $Q_{1}$ and $Q_{2}$ are moving to the right with $|\dot{Q}_{1}|=O(1)$, $|\dot{Q}_{2}|=O(\lambda_{1}^{-1/2})$ and the binary $Q_{3}$-$Q_{4}$ moves to the left with speed of order $O(1)$ by Proposition 3.6. After catching up and experiencing a two-body interaction with $Q_{2}$, the particle $Q_{1}$ would be reflected and would get close to the binary, reaching the section $\mathcal{S}_{-}$, on which, by Proposition 3.8, the distance between $Q_{1}$ and $Q_{2}$ is $\chi_{1}=O(f_{\mu,\lambda_{1}}\lambda_{1}\chi_{0}),$ and the time $t_{1}$ that (after the renormalization) the orbit spend between the sections $\mathcal{S}_{+}$ and $\mathcal{S}_{-}$, is $t_{1}=O(f_{\mu,\lambda_{1}}\lambda_{1}\chi_{0}).$

In the original spacetime scale without the renormalization, the distance between $Q_{1}$ and $Q_{2}$ is $\bar{\chi}_{1}=O(f_{\mu,\lambda_{1}}\chi_{0}),$ and the traveling time is $\bar{t}_{1}=\begin{cases}&O(\mu^{-1/2}\chi_{0}),\mathrm{\ if\ }\lambda_{1}=\lambda_{*},\\ &O((\lambda_{*}/\lambda_{1})^{1/2}\chi_{0}),\mathrm{\ if\ }\lambda_{1}\geq 2\lambda_{*}.\end{cases}$ Then $Q_{1}$ and the binary $Q_{3}$-$Q_{4}$ would come to another near triple collision, after which the second renormalization $\mathcal{R}_{\lambda_{2}}$ is performed. We need to update $\chi_{0}$ to $\chi_{1}$ and $\lambda_{1}$ to $\lambda_{2}$ for estimating $\chi_{2}$ and $t_{2}$ and update $\lambda_{1}$ to $\lambda_{1}\lambda_{2}$ for estimating $\bar{\chi}_{2}$ and $\bar{t}_{2}$, and repeat the argument.

We then perform the induction. Suppose that during the first $i$-steps, there are $j(i)$ times with $\lambda_{k}=\lambda_{*}$, $1\leq k\leq i$. Then we get from Proposition 3.8, we see that in the original spacetime scale, the timespan between the $i$-th renormalization $\mathcal{R}_{\lambda_{i}}$ and the $(i+1)$-th renormalization $\mathcal{R}_{\lambda_{i+1}}$ is

[TABLE]

and the distance $\bar{\chi}_{i}=O(\mu^{-j(i)/2}\mu^{\dagger}\lambda_{*}^{i/2}\chi_{0}),$ where $\dagger=-\frac{1}{2}$ if $\lambda_{i}=\lambda_{*}$ and $\dagger=0$ if $\lambda_{i}\geq 2\lambda_{*}$.

Therefore the total timespan before the $(i+1)$-th renormalization is estimated as

[TABLE]

So, we have the followings two cases:

1. i)

if $\lim_{i\to\infty}T_{i}<+\infty$, then we obtain non-collision singularities; 2. ii)

If $\lim_{i\to\infty}T_{i}=+\infty$, then we have super-hyperbolic orbits.

For i), an easy choice is to take $\lambda_{k}>2\lambda_{*}$ for all $k=1,2,\ldots.$ Thus we have $j(k)=0$ for all $k=1,2,\ldots$. We thus see that the sequence $T_{i}$ converges exponentially, since $O$ is uniform. We may also take $\lambda_{k}\to\infty$. The velocity of the particle $Q_{1}$ is estimated as $O(\sqrt{\prod\lambda_{i}})$. In this way, we make $\frac{|\dot{Q}_{1}(t_{k+2})|}{|\dot{Q}_{1}(t_{k})|}\to\infty.$ This gives the statement ii) in Theorem 1.3.

For ii), the choice of $\{\delta_{i}\}$ is more subtle. We give two valid selections here. The first one is to take $\delta_{i}=\delta_{*}$, hence, $\lambda_{i}=\lambda_{*}$. Then by (3.9), $T_{i}=O(\sum_{k=1}^{i}\mu^{-k/2}\chi_{0})\to\infty$ as $i\to\infty$ and since at time $T_{i}$ the distance between $Q_{1}$ and $Q_{2}$ is $O(\mu^{-i/2}\lambda_{*}^{i/2}\chi_{0})$, we have

[TABLE]

In fact this is the slowest rate of growths for the super-hyperbolic orbits constructed here.

A second choice of $\{\delta_{i}\}$ is of the form $\delta_{i}=\mu^{N_{i}}\delta_{*}$, satisfying (3.7). Here $N_{i}\in\mathbb{N}$, $N_{1}\geqslant 1$ and if for some $i$, $N_{i}\geqslant 1$, then $N_{k}=0$, $k=i,\dots i+N_{i}$ and $N_{i+N_{i}+1}\geqslant 1$. Let $\{i_{k},k=1,\dots\}$ be the increasing sequence such that $N_{i_{k}}\geqslant 1$ and $N_{i}=0$ if $i\not\in\{i_{k}\}$. Then by (3.8), we get $\lim_{i\to\infty}T_{i}\to\infty$ since $T_{i}$ grows by a multiple of $\sqrt{\mu}$ if $i\not\in\{i_{k}\}$.

We thus prove the assertion of Theorem 1.3. ∎

Remark 3.14**.**

A straightforward analysis of the construction we can see that while the non-collision singularity is allowed for all $m_{2}>\frac{m_{1}^{2}+2m_{1}}{2-m_{2}}$, the super-hyperbolic orbit is possible only for $0<m_{2}-\frac{m_{1}^{2}+2m_{1}}{2-m_{2}}\ll 1$. The latter restriction is essential in our construction for slowing down the the speeds of the particles. In [SX], the existence of super-hyperbolic orbits was presented for a much wider range of mass ratios in the Mather-McGehee model of collinear four-body problem. Infinitely many double collisions are not avoidable there.

4. Estimate of the return time

In this section, we first write down the Hamiltonian of the F4BP and study the transformation between the left and right Jacobi coordinates in the first two subsections. After that, we give the proof of Proposition 3.8.

4.1. The Hamiltonian systems in Jacobi coordinates

The original Hamiltonian has the form

[TABLE]

In the (left) Jacobi coordinates, defined in Section 2.1, the Hamiltonian reads

[TABLE]

where $M_{0}=\frac{1}{2},\ M_{1}=\frac{2m_{1}}{m_{1}+2},\ M_{2}=\frac{m_{2}(m_{1}+2)}{m_{1}+m_{2}+2}.$ The first parenthesis is a three-body problem in Jacobi coordinates and the second one is a perturbed two-body problem. We shall used this Hamiltonian when the orbits of the system lie between the two sections $\mathcal{S}_{-}$ and $\mathcal{S}_{+}$ and close to triple collision.

When $Q_{1}$ is far away from the binary and $Q_{2}$, i.e. the orbit has not reached the section $\mathcal{S}_{-}$ and have exited the section $\mathcal{S}_{+}$, we choose a different way of grouping terms and use the following form of Hamiltonian system, which is a perturbation of three Kepler problems

[TABLE]

We denote the energy

[TABLE]

for the Kepler problem $(\mathbf{x}_{i},\mathbf{p}_{i})$, $i=0,1,2.$

When $Q_{1}$ is much closer to $Q_{2}$ than the binary $Q_{3}$-$Q_{4}$, we use the right Jacobi coordinates.

[TABLE]

Here, $(\mathbf{x}_{0}^{R},\mathbf{p}_{0}^{R})$ is the same as $(\mathbf{x}_{0},\mathbf{p}_{0})$, the variables $(\mathbf{x}_{1}^{R},\mathbf{p}_{1}^{R})$ describes the relative motion between $Q_{1}$ and $Q_{2}$, and $(\mathbf{x}_{2}^{R},\mathbf{p}_{2}^{R})$ does that between the mass center of $Q_{1}$-$Q_{2}$ and the mass center of the binary $Q_{3}$-$Q_{4}$.

For the above used left Jacobi coordinates (2.1) as well as the constants $M_{i},k_{i},m_{i}$, we shall put a superscript $L$ to avoid confusion with the right Jacobi coordinates (4.4) and relevant constants below. However, if there is no danger of confusion, we omit the superscripts.

To decide which set of coordinates to use, we introduce a middle section $\mathcal{S}^{M}=\{|\mathbf{x}_{1}^{R}|=\beta\chi\}$ where $\beta=\lambda^{-1/2}$ and $\lambda$ is the constant defining the renormalization. To the left (respectively right) of the section $\mathcal{S}^{M}$, we use the left (respectively right) Jacobi coordinates.

In the right case, $Q_{1}$ and $Q_{2}$ form a close pair and they are far away from the pair $Q_{3}$-$Q_{4}$, we write the Hamiltonian as the following

[TABLE]

Note that this is a system of three almost decoupled Kepler problems perturbed by the potential $U_{12}$.

4.2. Transition from the left to the right

We now give the linear transform changing the left Jacobi coordinates to the right Jacobi coordinates and vise versa. Let us denote $(\mathbf{x}_{0}^{L},\mathbf{x}_{1}^{L},\mathbf{x}_{2}^{L})^{T}=\mathcal{L}(q_{3},q_{1},q_{2})^{T}$,  $(\mathbf{x}_{0}^{R},\mathbf{x}_{1}^{R},\mathbf{x}_{2}^{R})^{T}=\mathcal{R}(q_{3},q_{1},q_{2})^{T}$, $(\mathbf{p}_{0}^{L},\mathbf{p}_{1}^{L},\mathbf{p}_{2}^{L})=\mathcal{L}^{-T}(P_{3},P_{1},P_{2})^{T}$, and $(\mathbf{p}_{0}^{R},\mathbf{p}_{1}^{R},\mathbf{p}_{2}^{R})=\mathcal{R}^{-T}(P_{3},P_{1},P_{2})^{T}$, where $-T$ means transpose inverse. Then

[TABLE]

Therefore,

[TABLE]

Hence we have the following explicit formulas for the transitions of coordinates,

[TABLE]

[TABLE]

and

[TABLE]

4.3. Proof of Proposition 3.8

We consider the scenario where the triple $Q_{1}$-$Q_{3}$-$Q_{4}$ is ejected from the triple collision such that $|\mathbf{p}_{0}|$ and $|\mathbf{p}_{1}|$ are sufficiently large and $|\mathbf{p}_{2}|\sim 1$. Suppose the distance between $Q_{1}$ and $Q_{2}$ is $\chi$, which is as large as we wish. After renormalization $\mathcal{R}_{\lambda}$, $\mathbf{p}_{0}$ and $\mathbf{p}_{1}$ are rescaled to size of order $O(1)$ and $|\mathbf{p}_{2}|=O(\beta)$, where we denote $\beta=\lambda^{-1/2}$. Then $Q_{1}$ would catch up with $Q_{2}$. For orbits traveling between sections $\mathcal{S}_{\pm}$ and $\mathcal{S}^{M}$, we have the following lemma.

Lemma 4.1**.**

There exists $C>1$ such that

1. (1)

for the I4BP orbits $\gamma(t):[0,T]\to\mathcal{M}$ with $\gamma(0)\in\mathcal{S}_{+}$ and $\gamma(T)\in\mathcal{S}^{M}$ such that

[TABLE]

we have $T=O(\chi)$ and

[TABLE] 2. (2)

for I4BP orbits $\gamma(t):[0,T]\to\mathcal{M}$ with $\gamma(0)\in\mathcal{S}^{M}$ and $\gamma(T)\in\mathcal{S}^{M}$ such that along $\gamma$ there exists a double collision between $Q_{1}$ and $Q_{2}$, and

[TABLE]

we have $T=O(\beta\chi)$ and $\max_{t\in[0,T]}\{|L_{i}^{R}(t)-L^{R}_{i}(0)|,i=0,1,2\}=O(\frac{1}{\chi}).$

We refer the reader to Lemma 6.6 of [X1] for a proof. The equations of motion are given in (B.3). The lemma follows from integrating equation (B.3) combined with a simple bootstrap argument. The argument goes as follows. We start by assuming the oscillations of $L_{i},\ i=0,1,2$ are bounded by a generous constant $C\gg\epsilon^{2}$. Then this implies that $r_{0}$ is bounded, and $r_{1}$ and $\ell_{1}$ grows at least linearly in time. Then integrating the estimates in Appendix (B.3) over time of order $\chi$, we get that the oscillations of $L_{i}$ are indeed $O(\epsilon^{2})$, which is the value of $(\ell_{1}^{i})^{-2}=\int_{\ell_{1}^{i}}^{\chi}\frac{1}{\ell_{1}^{3}}$ evaluated on the section $\mathcal{S}_{+}$. The same argument works for statement (2) with the estimates (B.8) in Appendix B.

We are now ready to prove Proposition 3.8.

Proof of Proposition 3.8.

By assumption, after the renormalization $\mathcal{R}_{\lambda_{\delta}}$, $\lambda=\lambda(\delta)$, on the section $\mathcal{S}_{+}$, we have

[TABLE]

By Lemma 4.1, we know that when the orbits reach the middle section $\mathcal{S}^{M}:=\{|\mathbf{x}_{1}^{R}|=\beta\chi\}$, we have

[TABLE]

On the section $\mathcal{S}^{M}$, using (4.7), we change from the left Jacobi coordinates to the right Jacobi coordinates and have $\mathbf{p}_{1}^{R}=O(1)$ and $\mathbf{p}_{2}^{R}=O(1)$. Both are pointing to the right. Then by Lemma 4.1, when the orbits return to the section $\mathcal{S}^{M}$, the values $|\mathbf{p}_{1}^{R}|$ and $|\mathbf{p}_{2}^{R}|$ undergoes an oscillation of order $O(\epsilon^{2})$ with $\mathbf{p}_{1}^{R}$ changing direction from right to left. Switching back to the left Jacobi coordinates, using (4.7) and (4.6), we then obtain on the section $\mathcal{S}^{M}$,

[TABLE]

where $\mathbf{p}_{1}^{L}$ and $\mathbf{p}_{2}^{L}$ are as in (4.8) up to an order $O(\epsilon^{2})$ oscillation). Also note that now we have $|\mathbf{x}_{1}^{L}|=O(\chi)$ and by assumption $\epsilon<\mu$. Therefore we have on the section $\mathcal{S}^{M}$, $|\bar{\mathbf{p}}_{2}^{L}|\in\frac{2m_{2}\sqrt{2M_{1}^{L}}}{m_{1}+m_{2}}(1-2\mu,1+2\mu)$ with direction pointing to the right and $\bar{\mathbf{p}}_{1}^{L}$ points to the left (thus we have $m_{1}(m_{1}+m_{2}+2)-2m_{2}<0$) with length

[TABLE]

Now we have two cases from the assumption:

1. (1)

$\lambda=\lambda_{*}$ and $m_{2}$ is $(m_{1},\mu,\lambda_{*})$-admissible, 2. (2)

$\lambda\geq 2\lambda_{*}$ and $m_{2}$ is $(m_{1},\mu,\lambda_{*})$-admissible, but not $(m_{1},\mu,\lambda)$-admissible.

In case (1), by equation (3.3)), we get that the first row of the above $|\bar{\mathbf{p}}_{1}^{L}|$ lies in the interval $[1,10]\cdot\sqrt{\mu\lambda_{*}^{-1}}\sqrt{2M_{1}^{L}}$. Note also that (3.3) implies that $\frac{m_{1}(m_{1}+m_{2}+2)-2m_{2}}{(m_{1}+2)(m_{1}+m_{2})}=O(\lambda_{*}^{-1/2})$, thus we get the second row of the above $|\bar{\mathbf{p}}_{1}^{L}|$ is estimated as $O(\mu\lambda_{*}^{-1/2})$. Thus we get $|\bar{\mathbf{p}}_{1}^{L}|\in[\frac{1}{2},11]\cdot\mu^{1/2}\lambda_{*}^{-1/2}\sqrt{2M_{1}^{L}}$.

In case (2), we have that $\frac{2m_{2}-m_{1}(m_{1}+m_{2}+2)}{(m_{1}+2)(m_{1}+m_{2})}$ is $O(\sqrt{\mu\lambda^{-1}})$ close to $\lambda_{*}^{-1/2}\frac{8m_{1}m_{2}(m_{1}+m_{2}+2)}{(m_{1}+2)^{2}(m_{1}+m_{2})^{2}}$ since $m_{2}$ is $(m_{1},\mu,\lambda_{*})$-admissible. Thus we get that $|\bar{\mathbf{p}}_{1}^{L}|$ is of order $\lambda_{*}^{-1/2}$ for all $\lambda>2\lambda_{*}$.

In both cases, with the estimate of the velocity, we get the corresponding estimate of time spans and distances. The assertion of the proposition now follows. ∎

5. The full four-body problem and proof of Theorem 1.4

In this section, we prove Theorem 1.4 for the full four-body problem(F4BP). The definitions of the sections $\mathcal{S}_{\pm}$, the local, global and renormalization maps for the F4BP are similar to the isosceles case. We will show in this section that the super-hyperbolic orbit of the F4BP is obtained by slightly perturbing the I4BP orbit from Theorem 1.3. The main issue is to control the extra variables $(G_{0},g_{0},G_{1},g_{1},G_{2},g_{2})$ as well as excluding the possibility of collisions.

5.1. The coordinates and the diagonal form

For each pair of Jacobi coordinates $(\mathbf{x}_{i},\mathbf{p}_{i}),\ i=0,1,2,$ we convert it into Delaunay coordinates $(L_{i},\ell_{i},G_{i},g_{i})$. We always assume the total angular momentum is vanishing, i.e. $G_{0}+G_{1}+G_{2}=0.$ Since the Hamiltonian does not depend on each individual angle $g_{i}$ but depends only on the relative angles, this reduces the number of degrees of freedom by 1. By further restricting to the zero energy level and to the Poincaré sections, we can reduce two more variables when defining the local and global maps as we did in the last section. We define the Poincaré sections $\mathcal{S}_{\pm}$ in the same way as Definition 3.1. The variables to be removed are chosen to be $L_{1}$ and $\ell_{1}$ on the sections $\mathcal{S}_{\pm}$. Thus we have the following set of variables to describe the dynamics

[TABLE]

the variables on the sections $\mathcal{S}_{\pm}$. We shall estimate the matrices $d\mathbb{G}=\frac{\partial(X,Y)|_{\mathcal{S}_{-}}}{\partial(X,Y)|_{\mathcal{S}_{+}}}$, and $\ d\mathbb{L}=\frac{\partial(X,Y)|_{\mathcal{S}_{+}}}{\partial(X,Y)|_{\mathcal{S}_{-}}}$. As we state in the next lemma, the derivative matrix has a product structure in the isosceles limit, i.e. we evaluate the derivative matrix in the limit $|Y|\to 0$ (along an I4BP orbit).

Lemma 5.1**.**

The derivative matrix $\frac{\partial(X,Y)|_{\mathcal{S}_{-}}}{\partial(X,Y)|_{\mathcal{S}_{+}}}$ evaluated along an I4BP orbit has the following structure

[TABLE]

The same diagonal form of the derivative matrix also holds for the local map hence for the Poincaré return map.

We refer the reader to [GHX, Section 9.1] for a proof of this statement.

5.2. The I3BP near infinity

We will need the following special orbits $\gamma_{I}$ and $\gamma_{O}$ as guides of our orbits when experiencing near triple collisions. Follow our discussion before (see Proposition 3.8), after the two-body interaction with $Q_{2}$, we need that the relative speed between $Q_{1}$ and the binary $Q_{3}$-$Q_{4}$ is of order $\lambda_{*}^{-1/2}$. Suppose the triple $Q_{1}$-$Q_{3}$-$Q_{4}$ goes to triple collision, then we expect that for $\lambda_{*}$ sufficiently large, the orbit approaching triple collision stays very close to certain orbit $\gamma_{I}$ on the stable manifold $W^{s}_{\therefore}(O)$ with the feature that it comes from the infinity with almost zero initial velocity of $Q_{1}$. Therefore, to guide our incoming orbits of near triple collision, we choose the orbit $\gamma_{I}$ to be an orbit on $W^{s}_{\therefore}(O)$ approaching infinity parabolically as $t\to-\infty$, whose existence has been well studied (c.f. [SM]).

Proposition 5.2** ([SM]).**

There exists an orbit of the I3BP $Q_{1}$-$Q_{3}$-$Q_{4}$ lying on the stable manifold $W_{\therefore}^{s}(O)$ of the Lagrange fixed point $O$ and satisfying the boundary condition $\mathbf{x}^{L}_{1}(t)\to\infty$ and $\mathbf{p}^{L}_{1}(t)\to 0$ as $t\to-\infty$.

Next, if the triple $Q_{1}$-$Q_{3}$-$Q_{4}$ stays close to the right lower Lagrange fixed point for a long time and escapes to the right upper arm shadowing the unstable manifold on $\mathcal{M}_{0}$, we expect $r$ remains small for a long time. So we use $\gamma_{O}$, that is, the unstable manifold of the right lower Lagrange fixed point on the collision manifold $\mathcal{M}_{0}$, to guide the orbits leaving the triple collision.

Notation 5.3**.**

1. (1)

We will call the orbit in Proposition 5.2 $\gamma_{I}$ and the unstable manifold $W^{u}_{0}$ in Notation 2.3 $\gamma_{O}$. 2. (2)

The orbits $\gamma_{I}$ and $\gamma_{O}$ are invariant under time translation, thus for convenience, for $\gamma_{I}$, we assume $\gamma_{I}(0)\in\mathcal{S}_{-}$ and for $\gamma_{O}$ we assume $\gamma_{O}(0)\in\mathcal{S}_{+}$, after time translations if necessary.

Notation 5.4**.**

1. (1)

Denote by $\mathcal{M}$ the phase space of the F4BP. 2. (2)

Denote by $\mathcal{M}_{\therefore}$ the phase space of the three-body problem $($F3BP$)$ $Q_{1}$-$Q_{3}$-$Q_{4}$, which has six dimensions after reducing the momentum conservation and angular momentum conservation. 3. (3)

Denote by $\pi_{\therefore}$ the projection to the $\mathcal{M}_{\therefore}$ subsystem.

The next proposition shows that $\gamma_{I}$ indeed serves as a good guide for the incoming orbits.

Proposition 5.5** (Proposition 3.17 of [GHX]).**

Let $\delta^{\prime}$ and $\chi$ satisfy $\delta^{\prime}>\chi^{-D}$ for some constant $D>1$. Let $\gamma:\ [0,T]\to\mathcal{M}$ be an orbit of the I4BP such that

1. (1)

$|\mathbf{x}_{1}(\gamma(0))|\sim\chi$, $|\mathbf{p}_{1}(\gamma(0))|\leq\delta^{\prime}$; 2. (2)

$\gamma(T)\in\mathcal{S}_{-}$* satisfies $\mathrm{dist}(\pi_{\therefore}(\gamma(T)),W^{s}_{\therefore}(O))<\delta^{\prime}$.*

Then as $\delta^{\prime}\to 0$, we have $\mathrm{dist}(\pi_{\therefore}(\gamma(T)),\gamma_{I}(0))\to 0$.

5.3. The local map

In this section, we study the derivative of the local map. When the subsystem $Q_{1}$-$Q_{3}$-$Q_{4}$ is near triple collision, we shall treat the F4BP as a small perturbation of the product system of the F3BP and a Kepler problem $(\mathbf{x}_{2},\mathbf{p}_{2})$. Near triple collision of the $Q_{1}$-$Q_{3}$-$Q_{4}$ subsystem, we update the meanings of $X$ and $Y$ as $X=(r,v,\psi,w)$ the blowup coordinates that we used in the I3BP, and we introduce in addition the new scaling invariant variables $Y:=(w_{0}=r^{-1/2}G_{0},\mathsf{g}_{01},w_{2}=r^{-1/2}G_{2},\mathsf{g}_{21})$.

We first consider the F3BP $Q_{1}$-$Q_{3}$-$Q_{4}$. Assuming that the total angular momentum $G_{0}+G_{1}=0$, the system has three degrees of freedom, and we introduce the variables $w_{0},\mathsf{g}_{01}$ in addition to the blowup coordinates $(r,v,\psi,w)$ of the I3BP. The diagonal structure Lemma 5.1 also holds for the F3BP, so we can talk about the second diagonal block that is $2\times 2$ by projecting to the $(w_{0},\mathsf{g}_{01})$-plane.

Lemma 5.6**.**

When linearized at each Lagrange fixed point, the equations of motion of the F3BP projected to the $(w_{0},\mathsf{g}_{01})$-plane has one positive $\mu_{u}>0$ and one negative $\mu_{s}<0$ eigenvalues.

Notation 5.7**.**

Denote by $E_{3}^{s}(\gamma_{I}(\tau)),\ \tau\in[0,\infty)$ the tangent vector $($normalized to have length $1)$ of $W_{\therefore}^{s}(\gamma_{I}(\tau))$ at $(0,0)$ in the $(w_{0},\mathsf{g}_{01})$-plane $($for the local map$)$. Similarly, we define $E_{3}^{u}(\gamma_{O}(\tau))$, $\tau\in(-\infty,0]$.

We next consider the F4BP taking into further consideration of $Q_{2}$. Assuming the total angular momentum vanishes, we use coordinates $Y$ in addition to the blowup coordinates for the I3BP. Again by the diagonal structure Lemma 5.1, so we can talk about the second diagonal block by projecting to the $Y$-components. The following lemma describes the linear dynamics of the F4BP around the Lagrangian fixed point. Its proof is given in Proposition 8.2 of [GHX].

Lemma 5.8**.**

1. (1)

The linearized dynamics at the lower Lagrange fixed point, when projected to the $Y$-component, has four eigenvalues $\mu_{u}>-\frac{v_{*}}{2}>0>\mu_{s}$, where $\mu_{u}$ and $\mu_{s}$ are that in Lemma 5.6. Denote by $\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\in\mathbb{R}^{4}$ the eigenvectors respectively, then we have that the projection of $\mathbf{e}_{1}$ and $\mathbf{e}_{4}$ to the $(w_{0},\mathsf{g}_{01})$-plane are eigenvectors for the linearized dynamics in Lemma 5.6. Moreover, the vector $\mathbf{e}_{2}$ has nontrivial projection to the $G_{2}$ component, so does its pushforward along $\gamma_{O}$ by the tangent dynamics. 2. (2)

The tangent dynamics along $\gamma_{I}$ and $\gamma_{O}$ commutes with the projection to the $(w_{0},\mathsf{g}_{01})$-plane $($the F3BP$)$.

With the last lemma, we can construct the two-dimensional plane $E^{s}_{4}(\gamma_{I}(0))\subset T_{\gamma_{I}(0)}\mathcal{S}_{-}$ satisfying that its pushforward by the tangent flow along $\gamma_{I}(\tau)$ as $\tau\to\infty$ is $\mathrm{span}\{\mathbf{e}_{3},\mathbf{e}_{4}\}$. Similarly, we construct the 2 dimensional plane $E^{u}_{4}(\gamma_{O}(0))\subset T_{\gamma_{O}(0)}\mathcal{S}_{+}$ whose pushforward by the tangent flow along $\gamma_{O}(\tau)$ as $\tau\to-\infty$ is $\mathrm{span}\{\mathbf{e}_{1},\mathbf{e}_{2}\}$. Moreover, by Lemma 5.8, we have that the projection to the $(w_{0},\mathsf{g}_{01})$-component of the pushforward of $\mathbf{e}_{1}$ to $T_{\gamma_{I}(0)}\mathcal{S}_{-}$(respectively the pushforward of $\mathbf{e}_{4}$ to $T_{\gamma_{O}(0)}\mathcal{S}_{+}$) is $E^{s}_{3}(\gamma_{I}(0))$ (respectively $E^{u}_{3}(\gamma_{O}(0))$).

After the application of the local map, we perform a renormalization on the section $\mathcal{S}_{+}$. The definition of the renormalization is to extend the Definition 3.5 to the four-body problem. The renormalization does not change the values of the variables $(w_{0},\mathsf{g}_{01},w_{2},\mathsf{g}_{21})$, but we need to use the variables $(G_{0},\mathsf{g}_{01},G_{2},\mathsf{g}_{21})$ in order to apply the global map.

In the following propositions, without danger of confusions, when talking about the vectors $\mathbf{u}_{1},\mathbf{l}_{1}$ that were defined in Proposition 3.9 for the I4BP, we naturally embed them into the phase space of the F4BP by adding zeros to the extra dimensions. Similarly for the subspaces $E^{u/s}_{4}$, $E^{u/s}_{3}$, etc. Moreover, though the basepoint of the subspaces $E^{s}_{4}$, $E^{s}_{3}$ is $\gamma_{I}(0)$, we naturally parallel transport these subspaces to nearby points of $\gamma_{I}(0)$. Similarly for $E^{u}_{4}$, $E^{u}_{3}$.

Proposition 5.9**.**

For all $\eta>0$, there exists a small number $d_{0}>0$ such that the following holds: Let $\gamma:\ [0,T]\to\mathcal{M}$ be an orbit of the F4BP in the blowup coordinates with $\gamma(0)\in\mathcal{S}_{-},\ \gamma(T)\in\mathcal{S}_{+}$ and $\gamma(0)$ is sufficiently close to $\gamma_{I}(0)$ such that $|Y|<d_{0}e^{-\mu_{u}T},\ \chi>d_{0}^{-1}e^{\mu_{u}T}$ along $\gamma$, then we have

[TABLE]

[TABLE]

where $\mathbf{u}_{1}$ and $\mathbf{l}_{1}$ are as in Proposition 3.9. Moreover, for all $v\in P$, we have

[TABLE]

5.4. The global map

In this section, we study the derivative of the global map.

Proposition 5.10**.**

For all $\eta>0$, there exist $\chi_{0},\ \epsilon_{0},\nu_{0}$ such that for all $\chi>\chi_{0},\epsilon<\epsilon_{0},\nu<\nu_{0}$, we have the following: Let $\gamma:\ [0,T]\to\mathcal{M}$ be an orbit of the F4BP with $\gamma(0)\in\mathcal{S}_{+}$, $\gamma(T)\in\mathcal{S}_{-}$ and $|Y|\leq\nu$ along the orbit. Then there exist vector fields $\bar{\mathbf{u}}_{i},\ i=1,2,3,$ and $\bar{\mathbf{l}}_{i}$, $i=1,2$, such that we have

[TABLE]

and

[TABLE]

and for each vector $v\in\mathcal{C}_{\eta}(\mathbf{u}_{1},E_{4}^{u}(\gamma_{O}(0)))$, we have

[TABLE]

where $\mathbf{u}_{1}$ is as in Proposition 3.9, and $\bar{\mathbf{u}}_{1}$ and $\bar{\mathbf{l}}_{1}$ are as in Proposition 3.10. These vectors have definite limits in the isosceles limit $Y\to 0$ and $\chi\to\infty.$ Moreover, we have $dG_{2}\cdot\bar{\mathbf{u}}_{2}\neq 0$.

We give a proof of this statement in Appendix C, which is slightly different from that of [GHX]. The proof involves a careful analysis of the structure of the $o(f_{\mu,\lambda}\cdot\chi)$ error term, since there is no expansion in the $\bar{\mathbf{u}}_{3}$ direction (this is why we have (5.3)), we have to show that the $o(f_{\lambda,\mu}\cdot\chi)$ error in $d\mathbb{G}$ does not spoil $\bar{\mathbf{u}}_{3}$. All the above vectors can be given explicitly. We shall be interested in controlling the plane $\mathrm{span}(\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3})$ being pushforward by the flow, of which, one direction is used to control the $(G_{0},\mathsf{g}_{01})$, i.e. the motion of be binary and the other is used to control $(G_{2},\mathsf{g}_{21})$, i.e. to make sure that the particle $Q_{1}$ comes to a near triple collision configuration with the binary $Q_{3}$-$Q_{4}$ after application of the global map.

5.5. The transversality condition

When we compose $\mathcal{R}\mathbb{L}$ and $\mathbb{G}$ to get the Poincaré map $\mathbb{P}=\mathbb{G}\mathcal{R}\mathbb{L}$, we hope that $d\mathbb{P}$ preserves the cone $\mathcal{C}_{2\eta}(\bar{\mathbf{u}}_{1},\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3})$ and expands each vector inside. For this purpose, we have to verify that $\mathrm{span}(\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3})$ satisfies (5.1) for a plane $P$, i.e. $\mathrm{span}\{\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3}\}\cap\mathcal{C}_{\eta}(E_{4}^{s}(\gamma_{I}(0)))=\{0\}.$

Let $\mathbf{e}_{3}^{\prime},\mathbf{e}_{4}^{\prime}$ be two vectors in $E_{4}^{s}(\gamma_{I}(0))$ spanning unit area, then the transversality condition is given explicitly as the nonvanishing of the determinant $\det(\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3},\mathbf{e}_{3}^{\prime},\mathbf{e}_{4}^{\prime})\neq 0$. We have the following lemma.

Lemma 5.11**.**

The determinant $\det(\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3},\mathbf{e}_{3}^{\prime},\mathbf{e}_{4}^{\prime})$ is an analytic function of the masses $m_{1},m_{2}$, so it is nonvanishing for generic masses if it is so for one choice of $(m_{1},m_{2})$. In particular, for such masses we have $\text{span}\{\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3}\}\cap\mathcal{C}_{\eta}(E^{s}_{4}(\gamma_{I}(0)))=\{0\}$ for some $\eta>0$.

Note that for the I3BP, we have $G_{0}=0$, and double collision between the binary occurs when $\psi=\pm\pi/2$. Thus to avoid double collision between the binary, it is enough to make sure that $G_{0}$ and $\psi$ do not vanish simultaneously, which is guaranteed by the following proposition.

Notation 5.12**.**

1. (1)

Denote by $\mathcal{B}_{O}\subset\mathbb{R}$ the set of all the instant $t$ where the $w_{0}$ or $G_{0}$ component of $E^{u}_{3}(\gamma_{O}(t))$ vanishes. Similarly, we define $\mathcal{B}_{I}$ to be that of $E^{s}_{3}(\gamma_{I}(t))$. 2. (2)

The sets $\Psi_{O}$ and $\Psi_{I}$ are, respectively, the set of all the instant $t$ when the $\psi$-component of $\gamma_{O}$ and $\gamma_{I}$ has absolute value $\pi/2$.

Proposition 5.13**.**

For the choices of masses $m_{1}=m_{3}=m_{4}=1$ the following transversality conditions are satisfied: The sets $\mathcal{B}_{O}$ and $\mathcal{B}_{I}$ have finitely many elements, independent of $\chi$, and moreover, there exists $c>0$, independent of $\chi$, such that

[TABLE]

We present a numeric verification of the above statements in Appendix D.

5.6. Proof of Theorem 1.4

In this section, we give the proof of Theorem 1.4.

We fix $(m_{1},m_{2})$ and $\lambda$ as in Theorem 1.3 and $\lambda_{i}$, $i=1,\dots$ as in (3.7). We shall construct a genuine super-hyperbolic orbits (or non-collision singularities) close to those obtained there.

Step 1. the choice of admissible cube.

We next introduce the notion of admissible cube. Let us fix a small number $\eta>0$ throughout the proof.

Definition 5.14**.**

We say that a three-dimensional cube $\mathsf{C}_{i}$ on $\mathcal{S}_{-}$ is admissible for the $i$-th iteration, if

1. (1)

$|Y|<\bar{\nu}_{i}:=e^{\mu_{s}\sum_{k<i}T_{k}}$* for each point $(X,Y)$ on $\mathsf{C}_{i}$;* 2. (2)

$T\mathsf{C}\subset\mathcal{C}_{\eta}(\bar{\mathbf{u}}_{1},\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3})$; 3. (3)

When parametrized by the variables $\ell_{0},G_{0},G_{2}$, we have that for each $(G_{0},G_{2})$, the $\ell_{0}$-curve intersects $W^{s}_{3}$ and has length at least $\epsilon^{2}$. For each $(\ell_{0},G_{2})$, the curve on the $(G_{0},\sf g_{01})$-plane has length between $\eta\bar{\nu}_{i}$ and $\bar{\nu}_{i}$, and its two endpoints are on the boundary of the cone $\mathcal{C}_{\eta}(E_{3}^{s}(\gamma_{I}(0)))$. Note that the curve does not intersect $(G_{0},\mathsf{g}_{01})=(0,0)$ for $\eta$ small and its tangent vectors satisfy the cone condition in (5.1). For each $(\ell_{0},G_{0})$, the curve on the $(G_{2},\mathsf{g}_{21})$-plane has $G_{2}$ coordinates in the interval $[-1,1]\eta\bar{\nu}_{i}.$

Step 2. *the local map and the point-deleting procedure. *

We pick an admissible cube $\mathsf{C}_{i}$, then apply the local map followed by the renormalization. By Proposition 5.9 and the transversality condition Lemma 5.11, we know that the plane span$\{\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3}\}$ gets expanded by $d\mathbb{L}$ by at least $\min\{e^{\mu_{u}T_{i}},e^{-\frac{v_{*}}{2}T_{i}}\}=e^{-\frac{v_{*}}{2}T_{i}}$ since we have $\mu_{u}>-\frac{v_{*}}{2}$ (c.f. Lemma 5.8). Thus, the projection to the $Y$-components of the cube $\mathsf{C}_{i}$ is strongly expanded by at least $e^{-T_{i}v_{*}/2}$ with $T_{i}=O(\log\lambda_{i})$.

We next examine the dynamics of the $(w_{0},\mathsf{g}_{01})$-components closely. The Lagrange fixed point has two eigenvalues $\mu_{u}(>0>)\mu_{s}$ when restricted to the $(w_{0},\mathsf{g}_{01})$-plane. By the cone condition in (5.1) and by the $\lambda$-lemma, a curve on $\sf C_{i}$ with constant $\ell_{0},G_{2}$ component will approach $W^{u}_{3}(\gamma_{O}(0))$ exponentially fast (by $e^{T_{i}\mu_{s}}$) and get stretched by $e^{T_{i}\mu_{u}}$. We keep only the part of the curve that stays within the cone $\mathcal{C}_{\eta}(E^{s}_{3}(\gamma_{I}(\tau))),\ \tau\geq 0$ and $\mathcal{C}_{\eta}(E^{u}_{3}(\gamma_{O}(\tau))),\ \tau\leq 0$, along the orbit. Therefore the remaining segment has length estimated as $e^{\mu_{s}T_{i}}$.

So we introduce a point-deleting procedure on $\mathcal{R}\mathbb{L}\mathsf{C}_{i}$ as follows

1. (1)

In the $X$-component, we keep only a subsegment resulting from Proposition 3.4. 2. (2)

In the $(G_{0},\mathsf{g}_{01})$-component, along the flow, we keep only the part that stays within the cone $\mathcal{C}_{\eta}(E^{u}_{3}(\gamma_{O}(\tau))),\ \tau<0$. 3. (3)

In the $G_{2}$-component, we keep only the part with $G_{2}\in[-1,1]\eta e^{\mu_{s}T_{i}}\bar{\nu}_{i}$.

After the point-deleting procedure we obtain a cube $\bar{\mathsf{C}}_{i}$ on $\mathcal{S}_{+}.$

Step 3. the perturbation of the nonisoscelesness to the $X$-variables.

We next show that the perturbation of the nonisoscelesness $|Y|\neq 0$ to the $X$-variables ($X=(r,v,w,\psi)$ in blowup coordinates) in the I4BP is negligible.

For this purpose, we need the following notion of sojourn time to give an upper bound on the time defining the local map.

Definition 5.15** (Sojourn time).**

Let $\{T_{n}\}$ be a sequence satisfying $T_{n}>C_{1}^{-1}\log\delta_{0}^{-1}$, where $\delta_{0}$ is as in Proposition 3.4. We say that $\{T_{n}\}$ is a sequence of sojourn times of type-$(c,C),\ c>0,C>0$ if it satisfies in addition

[TABLE]

In the following, we assume that the above sojourn time estimates are satisfied for each step of the local map. Let us denote by $X^{\prime}=F(X)$ the equation of motion for the I4BP in the blowup coordinates and by $\bar{X}^{\prime}=F(\bar{X})+\mathcal{E}$ the equation of the F4BP, where the error $\mathcal{E}$ is estimated as $O(Y^{2})$ (see [GHX, Appendix A]). Then denoting $\delta X=X-\bar{X}$, we get $(\delta X)^{\prime}=DF\delta X+\mathcal{E}$ and we estimate the divergence of the orbits

[TABLE]

by Gronwall, where $M$ bounds $DF$ which is estimated by $\mu_{u}$. By the point deleting process, we see that $Y^{2}(0)$ is estimated by $e^{2\mu_{s}(\sum_{i=1}^{n-1}T_{i})}$, inductively obtained from the last step if $n>1$. By our assumption on the sojourn time $\tau\leq T_{n}$ we thus get an upper bound $|\delta X(\tau)|\leq\bar{C}e^{-\bar{D}T_{n}}$ for $\tau\in[0,T_{n}]$ and $\bar{D}=-\frac{Mc+\mu_{s}}{c}\gg 1$ if we choose $c$ small. For the $r$-variable, we note that the $r$-equation is $r^{\prime}=rv$, which does not involve the $Y$-variables, thus the first row of $\mathcal{E}$ vanishes. For the other variables variables $(v,w,\psi)$ of $X$, most of the time the orbit stays close to the Lagrangian fixed point thus the estimate of the fundamental solution of the equation $\delta X^{\prime}=DF(X)\delta X$ is dominated by exponentiating $DF$ restricted at the Lagrangian fixed point for time $T_{n}$. When $\bar{D}$ is much larger than $|\mu_{u}|$ and $|\mu_{s}|$, we see that the $O(e^{-\bar{D}T_{n}})$-perturbation to the $X$-variables is negligible.

Step 4. *the global map and the iteration. *

After the point-deleting procedure during the local map, we obtain a cube $\bar{\mathsf{C}}_{i}$ on $\mathcal{S}_{+}$ whose tangent lies in the cone $\mathcal{C}_{\eta}(\mathbf{u}_{1},E^{u}_{4}(\gamma_{O}(0)))$. We next apply the global map. We use $d\mathbb{G}$ in Proposition 5.10 to control the difference between an orbit issued from $\bar{\mathsf{C}}_{i}$ and an orbit of the I4BP. The assumption $|Y|\leq\nu$ of Proposition 5.10 is verified by the following lemma whose proof could be found in [GHX, Section 9.5].

Lemma 5.16**.**

There exist $C>0$, $\tilde{C}>0$ and $\nu_{0}>0$ independent of $\chi$ such that

1. (1)

for the escaping piece of an orbit, if on $\mathcal{S}_{+}$,

[TABLE]

then we have

[TABLE] 2. (2)

for the returning piece of an orbit, if on $\mathcal{S}_{-}$,

[TABLE]

then we have

[TABLE]

Note that the point-deleting procedure during each local map decreases the value of $|Y|$ by $e^{\mu_{s}T_{i}}.$ Thus to verify $|Y|\leq\nu$ for the global map with the last lemma, it is enough to choose $T_{1}$ large so that $e^{\mu_{s}T_{1}}<\nu$. This can be done by choosing $\chi_{0}$ and $\lambda_{*}$ large.

The last lemma also implies that when projected to the $(G_{0},\mathsf{g}_{01})$ plane, the orbit never intersect the piont $(0,0)$. Next, estimate (5.3) in Proposition 5.10 implies that $\mathbb{G}(\bar{\mathsf{C}}_{i})$, when projected to the plane span$\{\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3}\}$, covers a disk of radius $\frac{1}{4}e^{\mu_{s}T_{i}}$ centered at zero. We perform another point-deleting procedure so that we get an admissible cube $\mathsf{C}_{i+1}$ from $\mathbb{P}({\mathsf{C}}_{i})$ according to item (3) of Definition 5.14.

Thus, we can iterate the procedure for infinitely many steps. After pulling back to the initial admissible cube $\mathsf{C}_{i}$ along the flow, the point-deleting procedure gives rise to a point-deleting procedure on the initial admissible cube $\mathbb{P}^{-i}(\mathsf{C}_{i})\subset\mathsf{C}_{0}$, hence after infinitely many steps, we get a Cantor set as initial conditions.

Step 5. *No binary collision. *

For the local map, it may happen that $G_{0}$ vanishes at some isolated points whose number does not depend on $\chi$ when near triple collision. However, when this happens, by Proposition 5.13, we have $\psi\neq\pm\pi/2$, so the binary collision actually does not occur for the local map piece of orbit.

We next consider the global map. First, by Lemma 5.16, for the whole orbit defined by the global map at step $i$, we have $G_{0}\in[(1-2\epsilon)\tilde{\nu},(1+2\epsilon)\tilde{\nu}]$ with $\tilde{\nu}\in[\eta\bar{\nu}_{i},\bar{\nu}_{i}]$, since by the point-deleting procedure during the local map, when projected to the $(G_{0},w_{01})$, we keep only the part within the cone $\mathcal{C}_{\eta}(E_{3}^{s}(\gamma_{O}(0)))$, which is almost parallel to $E^{u}_{3}(\gamma_{O}(0))$ by the $\lambda$-lemma. In particular, $G_{0}$ never vanishes, which implies that the double collision between the binary $Q_{3}$-$Q_{4}$ does not occur.

It remains to exclude the possibility of binary collision between $Q_{1}$ and $Q_{2}$. We invoke the following proposition whose proof can be found in Section 6.

Proposition 5.17**.**

There is no double collision between $Q_{1}$ and $Q_{2}$ for the global map piece of orbits returning to an admissible cube on $\mathcal{S}_{-}$.

To summarize, we have excluded all the possible collisions for an orbit issued from the Cantor set.

For any initial point in the Cantor set, the estimate of the time during each return is the same as that in Theorem 1.3, thus we get a super-hyperbolic orbit or a non-collision singularities depending on the choice of sequence $\{\lambda_{i}\}$. We hence prove Theorem 1.4. ∎

6. Excluding double collisions

We are now in position to prove Proposition 5.17 on excluding the double collision between $Q_{1}$ and $Q_{2}$.

6.1. Proof of Proposition 5.17

Proof of Proposition 5.17.

Suppose a piece of orbit $\gamma:\ [0,T]\to\mathcal{M}$ defining the global map is issued from the section $\mathcal{S}_{+}$ and returns to $\mathcal{S}_{-}$ and experiences a double collision between $Q_{1}$ and $Q_{2}$ at time $T^{\prime}$.

Let us suppose that after the near triple collision on the section $\mathcal{S}_{+}$, we have $G_{1}^{L}=\bar{\nu}$ that is almost the number $\bar{\nu}_{n}$ after the $n$-th step of renormalization by the above point-deleting procedure.

In the following, we shall show that

**Claim: **Suppose there is a double collision between $Q_{1}$ and $Q_{2}$, then there is a constant $c$ independent of $\chi$ and $\bar{\nu}$, such that we have $|G_{2}^{L}(T)|\geq c\bar{\nu}$ when the orbit comes back to the section $\mathcal{S}_{-}.$

Note that the conclusion of this claim does not meet the item (3) in our definition of admissible cube (c.f. Definition 5.14) if we choose $\eta$ sufficiently small therein. Thus our choice of admissible cubes excludes the possibility of collisions between $Q_{1}$ and $Q_{2}$ for any orbit issued from the Cantor set constructed in the last section.

We next give the proof of the claim. The heuristic idea is that a collision between $Q_{1}$ and $Q_{2}$ will transfer a big amount of angular momentum of $G_{1}^{L}$ to $G_{2}^{L}$, so we can bound $G_{2}^{L}$ from below. It is important to introduce a new section $\mathcal{S}_{d}:=\{|x_{1}^{R}|=d\chi\}$ for some sufficiently small $d$ to be determined below, which is a section $d\chi$-distance away from the double collision of $Q_{1}$-$Q_{2}$.

The section $\mathcal{S}_{d}$ break the piece of orbit $\gamma$ into three pieces:

1. (1)

The segment from $\mathcal{S}_{+}$ to first intersecting $\mathcal{S}_{d}$; 2. (2)

The segment from first intersecting $\mathcal{S}_{d}$ to second intersecting $\mathcal{S}_{d}$; 3. (3)

The segment from leaving $\mathcal{S}_{d}$ to $\mathcal{S}_{-}$.

For the first and third pieces, we shall work in the left Jacobi coordinates and show that $G_{1}^{L}$ and $G_{2}^{L}$ do not oscillate much, but for the second segment, we shall work in the right Jacobi coordinates and show that all the Delaunay coordinates do not oscillate much, but when converted to the left Jacobi coordinates, there would be a big amount of angular momentum transferred to $G_{2}^{L}$.

For the piece (1), we apply Lemma 5.16 to get that when arriving at the section $\mathcal{S}_{d}$, we have the estimate $|G_{0}|\geq\frac{9}{10}\bar{\nu}.$

We next consider piece (2). It would be convenient to introduce the angle $\alpha:=\angle(\mathbf{x}_{1}^{R},\mathbf{x}_{2}^{R})$ (see Figure 3) measured at the time when the orbit first cross the section $\mathcal{S}_{d}$.

We shall use the following lemma whose proof is postponed to the next subsection.

Lemma 6.1**.**

The angle $\alpha:=\angle(\mathbf{x}_{1}^{R},\mathbf{x}_{2}^{R})$ is estimated as $C^{-1}\frac{\bar{\nu}}{\chi}\leq|\alpha|\leq C\frac{\bar{\nu}}{\chi}$.

Collision occurs if $\mathbf{x}_{1}^{R}=0$. We let $\gamma_{in}$ be the piece of orbit coming to the collision with time reversed and let $\gamma_{out}$ be the piece of orbit ejected from the collision. After a time translation, we let $T_{in}$ (respectively $T_{out}$) be the time such that $\gamma_{in}$ (respectively $\gamma_{out}$) intersects the section $\mathcal{S}_{d}$ and [math] be the collision time. We shall compare the orbit parameters of the two orbits such as $\delta G^{R}_{1}=G^{R}_{1}(T_{out})-G^{R}_{1}(T_{in})$ and similarly we apply the $\delta$-notation to other variables. Note that at the collision time $t=0$, we have $\delta G^{R}_{1}(0)=0,\ \delta g^{R}_{1}(0)=0.$

We shall integrate the Hamiltonian equations derived from Hamiltonian (4.5) over time $d\chi$. Let us consider the $\dot{G}_{1}^{R}$ equation first. We have $\dot{G}_{1}^{R}=-\frac{\partial H}{\partial g^{R}_{1}}=-\frac{\partial U_{12}}{\partial g^{R}_{1}}$ where the potential $U_{12}$ is in (4.5). Note that by assumption we have $|\mathbf{x}_{0}^{R}|=O(1),\ |\mathbf{x}_{1}^{R}|=O(d\chi)$, $|\mathbf{x}_{2}^{R}|\sim\chi$ and $\alpha=\angle(\mathbf{x}_{1}^{R},\mathbf{x}_{2}^{R})=O(\frac{\bar{\nu}}{\chi})$ by Lemma 6.1. Thus we estimate

[TABLE]

For both $\gamma_{in}$ and $\gamma_{out}$, we shall integrate the equation over time of order $d\chi$ to go from the boundary of $\{|\mathbf{x}_{1}^{R}|\leq d\chi\}$to the double collision and back. This gives the estimate $G_{1}^{R}(T_{in}),G_{1}^{R}(T_{out})=O(\frac{d^{2}\bar{\nu}}{\chi})$ since we have $G_{1}^{R}=0$ at collision.

By similar argument, we get

Lemma 6.2**.**

Suppose there is a collision between $Q_{1}$ and $Q_{2}$. Then we have the quantities $\delta G_{1}^{R},\delta G^{R}_{2},\delta g_{1}^{R}$ and $\delta g_{2}^{R}$ are all estimated by $O(\frac{d^{2}\bar{\nu}}{\chi})$.

By definition of $\alpha$, we have $(g_{1}-g_{2})^{R}(T_{in})=\alpha$, thus we have $(g_{1}-g_{2})^{R}(T_{out})=(1+O(d))\alpha$.

On the section $\mathcal{S}_{d}$, we convert to the left Jacobi coordinates. From (4.6), we get

[TABLE]

where $c_{i},\ i=1,2,3,4$ are positive numbers whose values are not important for us. This gives

[TABLE]

The first two terms on the RHS are estimated in the last lemma. We shall analyze the two latter terms in more details. It turns out that the main contribution is given by the term $\delta(\mathbf{x}^{R}_{2}\times\mathbf{p}_{1}^{R})$. We have the following lemma, whose proof is postponed to the next subsection.

Lemma 6.3**.**

Restricted to the section $\mathcal{S}_{d}$, we have $|\delta G^{L}_{2}|\geq c\bar{\nu}$ for some constant $c>0$ that is independent of $\bar{\nu}$ or $\chi$.

Finally, we consider piece (3). After leaving the section $\mathcal{S}_{d}$, we shall use the left Jacobi coordinates. We next consider the evolution of $G_{2}^{L}$ derived from the Hamiltonian (4.2). From the Hamiltonian equation, we get $\dot{G}_{2}^{L}=-\frac{\partial H}{\partial g_{2}^{L}}=-\frac{\partial U_{2}}{\partial g_{2}^{L}}$. To estimate the RHS of the equation, we note that

[TABLE]

This gives the estimate $|\dot{G}_{2}^{L}|\leq C\frac{|\mathbf{x}_{1}^{L}||\mathbf{x}_{2}^{L}|1/\chi}{|\mathbf{x}_{2}^{L}-\mathbf{x}_{1}^{L}|^{3}}=C\frac{\chi}{d^{3}\chi^{3}}$. Integrating over time of order $\mu^{-1/2}\lambda_{*}^{1/2}\chi$ by Proposition 3.8 using the fact that $|Q_{1}(t)-Q_{2}(t)|$ like $\mu^{1/2}\lambda_{*}^{-1/2}t$, we get

[TABLE]

where $T^{\prime}$ is a final time when $Q_{1}$ is close to the binary. To proceed, we need the following Sublemma whose proof is postponed to the next subsection

Sublemma 6.4**.**

For $m_{1}$ in a neighborhood of 1, we have the estimate $-\mu_{s}<-v_{*}$ for the I3BP.

Proof.

These quantities can be explicitly computed using the formulas from [GHX, Lemma D.1]. For $m_{1}=1$, direct calculation gives $\mu_{s}=-0.7848$ and $v_{*}=-\sqrt{6}$, which verify the assertion. ∎

Note that $\chi=\chi_{0}\prod_{i<n}\lambda_{i}=\chi_{0}e^{-v_{*}\sum_{i<n}T_{i}}$ after the $n$-th step of renormalization and $\bar{\nu}=e^{\lambda_{s}\sum_{i<n}T_{i}}$. Thus, we get by the Sublemma that $1/\chi\ll\bar{\nu}$, hence for large $\chi_{0}$, the estimate (6.4) implies that $G_{2}^{L}(T^{\prime})$ remains of order $\bar{\nu}$ when $Q_{1}$ gets close to the binary. This completes the proof of the claim. ∎

6.2. Proof of the lemmas

It remains to prove the lemmas used above.

Proof of Lemma 6.1.

First, from the triangle formed by $\mathbf{x}_{1}^{L}$ and $\mathbf{x}_{2}^{R}$ in the figure, we know that the angle $\alpha\sim\frac{1}{d}\angle(\mathbf{x}_{1}^{L},\mathbf{x}_{2}^{R})$. Note also that the two angles $\alpha$ and $\angle(\mathbf{x}_{1}^{L},\mathbf{x}_{2}^{R})$ add up to $\angle(\mathbf{x}_{1}^{L},\mathbf{x}_{1}^{R})$, since the latter is the outer angle of the former two. Thus for small $d$, the angle $\alpha$ is almost the same as $\angle(\mathbf{x}_{1}^{L},\mathbf{x}_{1}^{R})$.

We next estimate $\angle(\mathbf{x}_{1}^{L},\mathbf{x}_{1}^{R})$. By triangle inequality, we have

[TABLE]

In the following we shall show that $|\angle(\mathbf{x}_{1}^{R},\mathbf{p}_{1}^{L})|\ll|\angle(\mathbf{x}_{1}^{L},\mathbf{p}_{1}^{L})|$, thus $\alpha=|\angle(\mathbf{x}_{1}^{L},\mathbf{x}_{1}^{R})|$ is almost $|\angle(\mathbf{x}_{1}^{L},\mathbf{p}_{1}^{L})|$, which is estimated as $C\bar{\nu}/\chi$ using the angular momentum $G^{L}_{1}\sim\bar{\nu}$.

For the term $|\angle(\mathbf{x}_{1}^{R},\mathbf{p}_{1}^{L})|$ on the RHS of (6.5), we further split

[TABLE]

where the constant $m$ is chosen according to (4.7) such that $-m\mathbf{p}_{1}^{R}+\mathbf{p}_{1}^{L}$ is a multiple of $\mathbf{p}_{2}^{L}$, thus

[TABLE]

From Figure 3, we have $|\angle(\mathbf{x}_{1}^{L},\mathbf{x}_{2}^{L})|\sim d|\angle(\mathbf{x}_{1}^{R},\mathbf{x}_{2}^{L})|<d\alpha$, which is much smaller than $\alpha$. Next, to estimate $|\angle(\mathbf{x}_{2}^{L},\mathbf{p}_{2}^{L})|$, we need an estimate of $G_{2}^{L}$ on the section $\mathcal{S}_{d}$. We shall show below that $G_{2}^{L}=O(1/(d^{3}\chi))$, which combining $\mathbf{x}_{2}^{L}=O(\chi),\ \mathbf{p}_{2}^{L}=O(\lambda^{-1/2})$ gives $|\angle(\mathbf{x}_{2}^{L},\mathbf{p}_{2}^{L})|\leq C\sqrt{\lambda}/(d^{3}\chi^{2})\ll\alpha$. It remains to estimate $G_{2}^{L}$. We have the same estimate as (6.4) for the oscillations of $G_{2}^{L}$ from the section $\mathcal{S}_{+}$ to $\mathcal{S}_{d}$. Thus to get that $G_{2}^{L}=O(1/(d^{3}\chi))$ on the section $\mathcal{S}_{d}$, it is enough to prove a similar bound on $\mathcal{S}_{+}$. This follows from the $\chi^{2}$-expansion in the estimate of $d\mathbb{G}$ in Proposition 5.10 and $dG_{2}\cdot\bar{\mathbf{u}}_{2}$ therein. Indeed, since by the construction of the admissible cube in the last section, we require $G_{2}^{L}$ to be of order $\bar{\nu}$ on the section $\mathcal{S}_{-}$ after application of $\mathbb{P}$, thus its value on the section $\mathcal{S}_{+}$ before applying $\mathbb{P}$ is of order $\bar{\nu}/\chi^{2}$ by the $\chi^{2}$-expansion of $d\mathbb{G}$. Thus we have proved the estimate $G_{2}^{L}=O(1/(d^{3}\chi))$ hence complete the estimate $|\angle(\mathbf{x}_{1}^{L},-m\mathbf{p}_{1}^{R}-\mathbf{p}_{1}^{L})|\ll\alpha.$

The angle $\angle(\mathbf{x}_{1}^{R},m\mathbf{p}_{1}^{R})|=|\angle(\mathbf{x}_{1}^{R},\mathbf{p}_{1}^{R})|$ on the RHS of (6.6) can be estimated using the angular momentum $G_{1}^{R}$ and the collision assumption. At the collision time we have $G_{1}^{R}=0$. Next assuming $\alpha<C\nu/\chi$, we have the estimate $\dot{G}_{1}^{R}=O(d\bar{\nu}/\chi^{2})$ by (6.1), which implies $G_{1}^{R}$ on the boundary of $\mathcal{S}_{d}$ is $O(d^{2}\bar{\nu}/\chi)$ by integrating over time $O(d\chi)$. Since $G_{1}^{R}=\mathbf{x}_{1}^{R}\times\mathbf{p}_{1}^{R}$, we get $|\angle(\mathbf{x}_{1}^{R},\mathbf{p}_{1}^{R})|<d\bar{\nu}/\chi^{2}\ll$ since $|\mathbf{x}_{1}^{R}|=d\chi$ and $\mathbf{p}_{1}^{R}=O(1)$. Since the estimate of $|\angle(\mathbf{x}_{1}^{R},\mathbf{p}_{1}^{R})|$ is much less than the $C\bar{\nu}/\chi$ estimate of $\alpha$, thus (6.1) is always valid and the above argument is consistent.

Thus we have proved that the term $|\angle(\mathbf{x}_{1}^{R},\mathbf{p}_{1}^{L})|$ much smaller than $\alpha,$ hence completes the proof.

∎

Proof of Lemma 6.3.

To estimate $\delta G^{L}_{2}$, we analyze the RHS of (6.3) term by term. We first have $\delta G_{1}^{R},\delta G_{2}^{R}=O(d^{2}\bar{\nu}/\chi)$ by Lemma 6.2.

For the remaining two terms, we next consider the term $\delta(\mathbf{x}^{R}_{2}\times\mathbf{p}_{1}^{R})=O(\bar{\nu})$ bounded away from zero and independent of $\chi$, which is the main contribution to $\delta G_{2}^{L}$. Indeed, we have that $\mathbf{p}_{1}^{R}$ has reversed the direction by $\pi+O(d^{2}\bar{\nu}/\chi)$, $|x_{2}^{R}|\sim\chi$ by the estimate of $\delta g_{1}^{R}$ in Lemma 6.3 and the direction of $\mathbf{x}_{2}^{R}$ has changed by $O(d^{2}\bar{\nu}/\chi)$ by the estimate of $\delta g_{2}^{R}$ in Lemma 6.3. Thus we have

[TABLE]

Moreover, we have the estimate of the angle

[TABLE]

for some constant $c>0$, where the $O(d^{2}\bar{\nu}/\chi)$ estimate of $\angle(\mathbf{x}_{1}^{R},\mathbf{p}_{1}^{R})$ comes from the estimate $G_{1}^{R}=O(d^{2}\bar{\nu}/\chi)$ preceding Lemma 6.2 due to the collision. Thus we prove the estimate $\delta(\mathbf{x}^{R}_{2}\times\mathbf{p}_{1}^{R})\geq c\bar{\nu}.$

Arguing similarly, we get $\delta(\mathbf{x}_{1}^{R}\times\mathbf{p}_{2}^{R})=O(d\bar{\nu})$. The estimate is different from $\delta(\mathbf{x}^{R}_{2}\times\mathbf{p}_{1}^{R})=O(\bar{\nu})$ is because $|\mathbf{x}_{1}^{R}|=O(d\chi)$ and $|\mathbf{x}_{2}^{R}|=O(\chi)$ for both the initial and final moments.

This completes the proof. ∎

Appendix A Delaunay coordinates

In this appendix, we give the relations between Cartesian coordinates and Delaunay coordinates. Consider the Kepler problem $H(P,Q)=\frac{|P|^{2}}{2m}-\frac{k}{|Q|},$ $(P,Q)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$.

A.1. Elliptic motion

We have the following relations which explain the physical and geometrical meaning of the Delaunay coordinates. $a=\frac{L^{2}}{mk}$ is the semimajor axis, $b=\frac{LG}{mk}$ is the semiminor axis, $E=-\frac{k}{2a}$ is the energy, $G=Q\times P$ is the angular momentum, and $e=\sqrt{1-\left(\frac{G}{L}\right)^{2}}$ is the eccentricity. Moreover, $g$ is the argument of apapsis and $\ell$ is the mean anomaly. We can relate $\ell$ to the polar angle $\psi$ through the equations $\tan\frac{\psi}{2}=\sqrt{\frac{1+e}{1-e}}\cdot\tan\frac{u}{2},\quad u-e\sin u=\ell.$ We also have Kepler’s law $\frac{a^{3}}{T^{2}}=\frac{k/m}{(2\pi)^{2}}$ which relates the semimajor axis $a$ and the period $T$ of the elliptic motion.

Denoting the body’s position by $Q=(q_{1},q_{2})$ and its momentum by $P=(p_{1},p_{2})$ we have the following formulas in the case $g=0$

[TABLE]

where $u$ and $\ell$ are related by $u-e\sin u=\ell$. This is an ellipse whose major axis lies on the horizontal axis, with one focus at the origin and the other focus on the negative horizontal axis. The body moves counterclockwise if $G>0$.

A.2. Hyperbolic motion

For Kepler hyperbolic motion we have similar expressions for the geometric and physical quantities

[TABLE]

as well as the parametrization of the orbit

[TABLE]

where $u$ and $\ell$ are related by

[TABLE]

This hyperbola is symmetric with respect to the $x$-axis, open to the right, and the particle moves counterclockwise on it when $u$ increases ($l$ decreases) in the case when minus the angular momentum $G=Q\times P>0$. The angle $g$ is defined to be the angle measured from the positive $x$-axis to the symmetric axis. There are two such angles that differ by $\pi$, depending on the orientation of the symmetric axis. In the general case, we need to rotate the $(q_{1},q_{2})$ and $(p_{1},p_{2})$ using the matrix \mathrm{Rot}(g)=\left[\begin{array}[]{cc}\cos g&-\sin g\\ \sin g&\cos g\end{array}\right], that is, we have the following formula for $\mathbf{x}=(q_{1},q_{2})$ and $\mathbf{p}=(p_{1},p_{2})$,

[TABLE]

Appendix B The derivative of the global map in the I4BP

In this section, we give the proof of Proposition 3.10. The proof is very similar to Proposition 3.11 of [GHX]. with the difference as follows. The estimate in [GHX] consists of two segments: from the left $\mathcal{S}_{+}$ to the middle section $\mathcal{S}^{M}$ (where we exchange the left and right Jacobi coordinates) and from $\mathcal{S}^{M}$ to the right $\mathcal{S}_{-}$ (the right to the left case is similar). However, the estimate here consists of three segments: from $\mathcal{S}_{+}$ to $\mathcal{S}^{M}$, from $\mathcal{S}^{M}$ to the double collision of $Q_{1}$ and $Q_{2}$ and back to $\mathcal{S}^{M}$, and from $\mathcal{S}^{M}$ to the next $\mathcal{S}^{+}$. It looks the situation here is slightly complicated, but the proof and the result are completely analogous to that of [GHX]. Indeed, all the derivative matrices composing $d\mathbb{L}$ was already computed in [GHX], only the way of composing them is different. In the following, we give the proof and refer readers to [GHX] for some detailed calculations.

We first recall a formula from [X1, Section 7] computing the derivative of the Poincaré map defined by cutting the flow $\mathcal{V}^{\prime}=\mathcal{F}(\mathcal{V},t)$ between the sections $S^{i}$ and $S^{f}.$ We use $\mathcal{V}^{i}$ to denote the values of variables $\mathcal{V}$ restricted on the initial section $S^{i}$, while $\mathcal{V}^{f}$ means values of $\mathcal{V}$ on the final section $S^{f}$, and we use $t^{i}$ to denote the initial time and $t^{f}$ the final time. We want to compute the derivative $\mathcal{D}:=\frac{D\mathcal{V}^{f}}{D\mathcal{V}^{i}}$ of the Poincaré map. We have

[TABLE]

Here the middle term $\frac{D\mathcal{V}(t^{f})}{D\mathcal{V}(t^{i})}$ is the fundamental solution to the variational equation $(\delta\mathcal{V})^{\prime}=\nabla_{\mathcal{V}}\mathcal{F}(\mathcal{V},t)\delta\mathcal{V}$ and the two terms $\left(\mathrm{Id}-\mathcal{F}(t^{f})\otimes\frac{Dt^{f}}{D\mathcal{V}^{f}}\right)^{-1}$ and $\left(\mathrm{Id}-\mathcal{F}(t^{i})\otimes\frac{Dt^{i}}{D\mathcal{V}^{i}}\right)$ are called boundary contributions taking into account the issue that different orbits take different times to travel between the two sections.

We only consider orbits from Proposition 3.8. Recall that we introduce a middle section $\mathcal{S}^{M}=\{|\mathbf{x}_{1}^{R}|=\beta\chi\}$ and divide the orbit into three pieces, the escaping piece (from $\mathcal{S}_{+}$ to $\mathcal{S}^{M}$), the intermediate piece (from $\mathcal{S}^{M}$ to $\mathcal{S}^{M}$) and the returning piece (from $\mathcal{S}^{M}$ to $\mathcal{S}_{-}$). For the escaping and returning pieces of the orbit, we use the left Jacobi coordinates and for the intermediate piece, we use the right Jacobi coordinates. For all the three pieces, we shall take $\ell_{1}$ to be the new time and eliminate the variable $L_{1}$ using the energy conservation. Thus the remaining $X$-variables are $(L_{0},\ell_{0},L_{2},\ell_{2}).$ The proof of the Proposition 3.10 consists the following three steps.

• •

Step 1: estimating the fundamental solutions to the variational equation of all the three pieces.

• •

Step 2: the boundary terms $\mathrm{Id}+JD_{X}H\otimes\nabla_{X}\ell_{1}$ for all the three pieces.

• •

Step 3: the coordinate change from left to right and from right to left on the middle section $\mathcal{S}^{M}$.

Step 1: the variational equation. The Hamiltonian written in the Delaunay coordinates for the I4BP in the left Jacobi coordinate system has the form (see (4.2))

[TABLE]

where $U_{2}$ and $U_{01}$ are the restriction of that in (4.2) to the isosceles case:

[TABLE]

Note that when $r_{1}\gg r_{0}$, we have $U_{01}=\frac{k_{1}}{r_{1}}(1-\frac{1}{(1+r_{0}^{2}/4r_{1}^{2})^{1/2}})=\frac{k_{1}r_{0}^{2}}{8r_{1}^{3}}+O(\frac{r_{0}^{4}}{r_{1}^{5}}).$ We have the following estimates for the Hamiltonian equations for the escaping piece of the oribts (see equation (7.5) of [GHX]),

[TABLE]

Due to Lemma 4.1, we learn that $0<\frac{1}{C^{\prime}}\leqslant|\dot{\ell}_{1}|\leqslant C^{\prime}$ for some suitable $C^{\prime}>1$. So we treat $\ell_{1}$ as the new time variable. Then we remove the equation for variable $L_{1}$ from the system of equations (B.3) and divide all the equations for the remaining variables $(L_{0},\ell_{0},L_{2},\ell_{2})$ by $\dot{\ell}_{1}$. Denote by $X=(L_{0},\ell_{0},L_{2},\ell_{2})$, by $D_{X}=(\frac{\partial}{\partial L_{0}},\frac{\partial}{\partial\ell_{0}},\frac{\partial}{\partial L_{2}},\frac{\partial}{\partial\ell_{2}})$ the partial derivarives and by $\nabla_{X}$ the covariant derivative, $\nabla_{X}=D_{X}+(D_{X}L_{1})\partial_{L_{1}}$, which takes into account of the dependence on $X$ through $L_{1}$. The Hamiltonian equation now has the form $\frac{dX}{d\ell_{1}}=\frac{1}{\dot{\ell}_{1}}JD_{X}H$ and the variational equation has the form (see equation (C.2) of [GHX])

[TABLE]

After a lengthy but direct calculation (see equation (C.6) of [GHX] for details), we get the following form of the coefficient matrix (using the fact $L_{0}=O(1),\ L_{1}=O(1)$ and $L_{2}=O(\beta^{-1})$),

[TABLE]

Using Picard iteration (which stabilizes quickly and two steps are enough, see equation (C.7) of [GHX]), we have the following estimate for the fundamental solution of the corresponding variational equation (see equation (C.8) of [GHX]),

[TABLE]

For the returning piece of the orbit going from $\mathcal{S}^{M}$ to $\mathcal{S}_{-}$, we have for almost the whole time $L_{0}=O(1),\ L_{2}=O(1)$ and $L_{1}=O(f_{\mu,\lambda})$ (c.f. Proposition 3.10 for the definition of $f_{\mu,\lambda}$), since $L_{1}\sim|\mathbf{p}_{1}|^{-1}$ and $|\mathbf{p}_{1}|$ is estimated in Proposition 3.8, hence $\dot{\ell_{1}}\sim\frac{1}{L_{1}^{3}}=O(f^{-3}_{\mu,\lambda})$. Thus, in (B.5), we replace $\beta$ by 1 since $L_{2}$ has changed from $O(\beta^{-1})$ to $O(1)$, and multiply an additional factor $O(f^{3}_{\mu,\lambda})$ due to dividing $\dot{\ell}_{1}$. Next, by equation (A.4), we get $|\mathbf{x}_{1}|\sim L_{1}^{2}\ell_{1}$, thus when $|\mathbf{x}_{1}|$ ranges in from $\epsilon^{-1}$ to $O(\chi)$, we get that $\ell_{1}$ ranges from $f^{-2}_{\mu,\lambda}\cdot\epsilon^{-1}$ to $f^{-2}_{\mu,\lambda}\cdot\chi$. Then integrating the variational equation over with respect to the time $\ell_{1}$, we have the following estimates for its fundamental solution,

[TABLE]

For the intermediate piece of the orbit going from $\mathcal{S}^{M}$ back to itself, we consider the right Jacobi coordinate system. In the Delaunay coordinates, the Hamiltonian function (4.5) for the I4BP reads

[TABLE]

where we have

[TABLE]

In particular, when $r_{2}\gg r_{1}$, we have $U_{12}=\frac{k_{2}^{R}}{r_{2}^{R}}-\frac{2m_{1}+2m_{2}}{(\frac{1}{4}r_{0}^{2}+(r_{2}^{R})^{2})^{1/2}}+O(\frac{(r_{1}^{R})^{2}}{(r_{2}^{R})^{3}}).$ We have the following estimates for the system of equations as $\chi\to\infty$

[TABLE]

Again we treat $\ell_{1}^{R}$ as the new time and remove $L_{1}^{R}$ from the equation. Then using (B.4) and repeating the calculations for (B.5), we have the following form for the coefficient matrix of the corresponding variational equation,

[TABLE]

The timespan of the intermediate piece of the orbit is of order $O(\beta\chi)$, therefore, we have the following estimate for the fundamental solution

[TABLE]

Step 2, the boundary contributions.

The estimate of the boundary contributions is almost identical to that of [GHX] Appendix C, Step 2. The boundary terms $(\mathrm{Id}+\nabla_{X}\ell_{1}\otimes JD_{X}H)$ are calculated on the sections $\mathcal{S}_{+}$, $\mathcal{S}_{-}$ and $\mathcal{S}^{M}$. For the middle section $\mathcal{S}^{M}$, we need to compute the boundary contribution before and after changing Jacobi coordinates from the left one to the right one, and vise versa. We first consider the left side of the section $\mathcal{S}^{M}=\{|\mathbf{x}_{1}^{R}|=\beta\chi\}$ for the escaping piece of the orbit. Recall that $X=(L_{0},\ell_{0},L_{2}^{L},\ell_{2}^{L})$. For the Hamiltonian equation, we have $JD_{X}H=O\left(\frac{1}{\chi^{3}},1,\frac{1}{(\beta\chi)^{2}},\beta^{3}\right).$ Using (4.7), We get $\mathbf{x}_{1}^{R}=\frac{2}{m_{1}+2}\mathbf{x}_{1}^{L}-\mathbf{x}_{2}^{L}=\beta\chi$ on $\mathcal{S}^{M}$. Converting the Cartesian variables to Delaunay variables using formula from [GHX, Lemma B.1] and taking derivatives, we have

[TABLE]

Then by the implicit function theorem, we get $D_{X}\ell_{1}^{L}=O(0,0,\beta\chi,\frac{1}{\beta^{2}}),\;\partial_{L_{1}^{L}}\ell_{1}^{L}\simeq\chi.$ So, from the left side of the section $\mathcal{S}^{M}$, we have the boundary contribution (see equation (C.10) of [GHX]),

[TABLE]

Similarly, for the returning piece of the orbit, on the left side of $\mathcal{S}^{M}$, we have $D_{X}\ell_{1}^{L}=O(0,0,\mu^{1/2}\beta\chi,\mu^{1/2}\beta)$ and $\partial_{L_{1}^{R}}\ell_{1}^{R}=O(\mu^{1/2}\beta\chi)$. Then the boundary contribution takes the form,

[TABLE]

Next, we work from the right side of the section $\mathcal{S}^{M}$. On the right side, $X=(L_{0},\ell_{0},L^{R}_{2},\ell^{R}_{2})$. For the Hamiltonian vector filed, we have $JD_{X}H=O\left(\frac{1}{\chi^{3}},1,\frac{1}{\chi^{2}},1\right).$ Once again we substitute the formulas from [GHX, Lemma B.1] and take differentiation. We obtain, $\frac{1}{M^{R}_{1}k^{R}_{1}}(2L_{1}^{R}\ell_{1}^{R}dL_{1}^{R}+(L_{1}^{R})^{2}d\ell_{1}^{R})=0.$ Using implicit function theorem we have $D_{X}\ell_{1}^{R}=(0,0,0,0),\;\partial_{L_{1}^{R}}\ell_{1}^{R}\sim\beta\chi$. Therefore, from the right side of the section $\mathcal{S}^{M}$, we have the boundary contribution which is valid for both the escaping piece and returning piece of orbits (see Step 2 of Appendix C of [GHX])

[TABLE]

Finally, we consider the boundary contributions on the sections $\mathcal{S}_{\pm}$. Both were wokred out in Step 2 of Appendix C of [GHX], and we cite the result here directly. On the section $\mathcal{S}_{+}$, we have the boundary contribution

[TABLE]

and on the section $\mathcal{S}_{-}=\{r=\epsilon^{-1}\}$, the boundary contribution is,

[TABLE]

**Step 3, transition from the left to the right. **

At the middle section $\mathcal{S}^{M}$, we change the Delaunay coordinates on the left first to the left Jacobi coordinates, next from the left Jacobi to the right Jacobi coordinates, and finally from the right Jacobi to Delaunay coordinates on the right, and vice versa. This was done in Step 3 of Appendix C of [GHX]. We have the following transition matrix on the section $\mathcal{S}^{M}$ (see equation (C.11) of [GHX])

[TABLE]

Similarly, for the returning piece leaving $\mathcal{S}^{M}$, we have the following transition matrix from the right Jacobi coordinates to the left one,

[TABLE]

After completing the three steps, using the formula (B.1), we multiply all the matrices together $(\mathcal{D}_{-}\mathcal{D}_{R}\mathcal{D}_{M}^{R})\mathcal{M}_{L}(\mathcal{D}_{M}^{I}\mathcal{D}_{I}\mathcal{D}_{M}^{I})\mathcal{M}_{R}(\mathcal{D}_{M}^{E}\mathcal{D}_{E}\mathcal{D}_{+})$ and obtain (3.6). Hence, we complete the proof of Proposition 3.10. ∎

Appendix C The derivative of the global map in the F4BP

In this section, we study the $C^{1}$ estimate of the global map in the F4BP by allowing the variables $G_{i},g_{i}$, $i=0,1,2$, to be non-vanishing. Again the proof is very similar to that of [GHX] and the difference is as discussed at the beginning of the last appendix.

We shall first work out the estimate of $d\mathbb{G}$ in the isosceles limit $Y\to 0$, then handle the nonisoscelesness $Y\neq 0$.

C.1. The $C^{1}$-dynamics of the variables $G_{i}$, $g_{i}$, $i=0,1,2$ in the isosceles limit

In this section, we compute the derivative of the global map for the variables $(G_{0},g_{0},G_{1},g_{1},G_{2},g_{2})$. As we have done in Appendix B, we break the orbit from $\mathcal{S}_{+}$ to $\mathcal{S}_{-}$ into three pieces by the middle section $\mathcal{S}^{M}$. For each piece, we apply equation (B.1) to compute the derivative, then we also compute the derivative of the coordinate change from left to right and right to left on the middle section $\mathcal{S}^{M}$. Since we have the Hamiltonian equations $\dot{Y}_{i}=JD_{Y_{i}}H=0$ along an orbit of the I4BP, where J=\left[\begin{array}[]{cc}0&-1\\ 1&0\end{array}\right] and $Y_{i}=(G_{i},g_{i}),\ i=0,1,2$, we get that the boundary contributions in (B.1) are all identity, so in the following, we only work on the fundamental solutions to the variational equation as well as the left-right transitions. We stress that the contribution from the latter is essential for the $(G_{1},g_{1},G_{2},g_{2})$-component.

We split the proof into two steps:

1. (1)

the fundamental solution of the variational equation, 2. (2)

the transition from left to right and right to left.

It turns out that the estimate in item (1) is $O(1)$ and the $O(\chi^{2})$ expansion in the statement is given by item (2).

For the orbits under consideration, they have parameters $L_{1}^{L}=O(1)$, $L_{2}^{L}=O(\lambda^{1/2})=O(\beta^{-1})$ for the escaping pieces, $L^{R}_{1}=O(1)$, $L^{R}_{2}=O(1)$ for the intermediate pieces and $L_{1}^{L}=O(f_{\mu,\lambda})$ (see Appendix C, Step 1), $L_{2}^{L}=O(1)$ for the returning pieces.

**Step 1, the variational equation. **

Consider the variational equation evaluated on orbits between $\mathcal{S}_{+}$ and $\mathcal{S}^{M}$: the escaping piece. Denoting $Y_{0}=(G_{0},g_{0})$, $Y_{1}=(G_{1},g_{1})$, $Y_{2}=(G_{2},g_{2})$ and $Y=(Y_{1},Y_{2})$, then the Hamiltonian equation can be written as $\dot{Y}_{0}=JD_{Y_{0}}Y_{0}$, $\dot{Y}_{1}=JD_{Y_{1}}H,\ \dot{Y}_{2}=JD_{Y_{2}}H$ and the variational equation can be written as

[TABLE]

By straightforward estimates we know that the contributions from the entries $JD^{2}_{Y_{0}Y_{i}}$, $i=0,1,2$, to the solution to the variational equation are negligible (see equation (9.3) of [GHX]). The main contribution comes from the block \left[\begin{array}[]{cc}JD_{Y_{1}}^{2}H&JD_{Y_{2}Y_{1}}^{2}H\\ JD_{Y_{1}Y_{2}}^{2}H&JD_{Y_{2}}^{2}H\end{array}\right] (see equation (9.2) of [GHX]).

Between the sections $\mathcal{S}_{+}$ and $\mathcal{S}^{M}$ the quantity $r_{1}$ increase almost linearly with respect to time from the order $\epsilon^{-1}$ to the order $\chi$. Then we integrate the equation (C.1) over a time interval of order $\chi$, to get the following estimate of the fundamental solution (see equation (9.8) of [GHX]),

[TABLE]

where the matrix $\mathcal{D}_{G}^{L}$ is the fundamental solution of the variational equation given by \left[\begin{array}[]{cc}JD_{Y_{1}}^{2}H&JD_{Y_{2}Y_{1}}^{2}H\\ JD_{Y_{1}Y_{2}}^{2}H&JD_{Y_{2}}^{2}H\end{array}\right] and it has some structure that enables us to work it out explicitly (see equation (9.5) and (9.6) of [GHX]).

Similarly, for the intermediate piece of the orbits from the section $\mathcal{S}^{M}$ to itself, we integrate the estimate of the variational equations in equation (9.2) of [GHX] over time of order $\beta\chi$ to get the following form for the fundamental solution of the corresponding variational equation,

[TABLE]

where $\mathcal{D}_{I}^{G}=\mathrm{Id}+O(\beta_{4\times 4})$.

Finally, it follows from similar calculations that for the returning piece of the orbits between the sections $\mathcal{S}^{M}$ and $\mathcal{S}_{-}$, integrating the estimate of the variational equations in equation (9.2) of [GHX] over time $\ell_{1}$ ranging from $f_{\mu,\lambda}^{-2}\epsilon^{-1}$ to $f_{\mu,\lambda}^{-2}\chi$ (see the derivation of (B.7)), we get the following estimates for its fundamental solution,

[TABLE]

where the leading terms in $\mathcal{D}_{G}^{R}$ are given by $f_{\mu,\lambda}\mathcal{D}_{G}^{L}.$

Step 2, Transition from the left to the right.

We next consider the coordinate changes on the section $\mathcal{S}^{M}$ induced by switching between left and right Jacobi coordinates. For the escaping orbit passing through $\mathcal{S}^{M}$, we shall first convert the $Y$-variables into the left Jacobi coordinates, next convert from the left to right Jacobi coordinates using (4.7), then convert the right Jacobi coordinates back to the $Y$ varriables. We reverse the procedure for the returning orbit passing through $\mathcal{S}^{M}$. It turns out that each of the transition gives us a factor $\chi$, that is why we have $O(\chi^{2})$ in the statement.

Lemma C.1**.**

We have the following derivative estimate on the section $\mathcal{S}^{M}=\{|\mathbf{x}_{1}^{R}|=\beta\chi\}$:

(1) When the escaping orbit arrives at $\mathcal{S}^{M}$, for the left-right transition, we have

[TABLE]

where $\mathbf{v}^{R}=(1,-\frac{1}{L_{1}^{R}},-1,\frac{1}{L_{2}^{R}})^{T}$ and $dG_{1}^{R}:=\frac{\partial G_{1}^{R}}{\partial Y^{L}}\simeq\chi\cdot(\frac{1}{L_{1}^{L}},1,-\frac{1}{L_{2}^{L}},-1)$ with $L_{1}^{L},L_{1}^{R},L_{2}^{R}=O(1)$ and $L_{2}^{L}=O(\beta^{-1})$;

(2) When the returning orbit leaving $\mathcal{S}^{M}$, for the right-left transition, we have (we add a bar to the variables to distinguish them from the last item)

[TABLE]

where $\mathbf{v}^{L}=(1,-\frac{1}{\bar{L}_{1}^{L}},-1,\frac{1}{\bar{L}_{2}^{L}})^{T}$ and $d\bar{G}_{1}^{L}:=\frac{\partial\bar{G}_{1}^{R}}{\partial\bar{Y}^{L}}\simeq\chi\cdot(-\frac{1}{\bar{L}_{1}^{R}},1,-\frac{1}{\bar{L}_{2}^{R}},-1)$ with $\bar{L}_{2}^{L},\bar{L}_{1}^{R},\bar{L}_{2}^{R}=O(1)$ and $\bar{L}_{1}^{L}=O(f_{\mu,\lambda})$.

For a proof of this lemma, we refer the reader to [GHX, Appendix E] where the statement (1) is proved and the proof of statement (2) is almost identical.

Hence, after combing (C.2)-(C.4) and Lemma C.1, the derivatives of the global map for the variables $G_{i},g_{i}$, $i=0,1,2$, between the sections $\mathcal{S}_{+}$ and $\mathcal{S}_{-}$ takes the following form,

[TABLE]

where $\bar{\mathbf{u}}_{2}=\mathcal{D}_{G}^{R}\mathbf{v}^{L}+O(\epsilon^{2})=(1,-\frac{1}{\bar{L}_{1}^{L}},-1,\frac{1}{\bar{L}_{2}^{L}})+O(\epsilon^{2})$ and $\bar{\mathbf{l}}_{2}=\frac{1}{\chi}dG_{1}^{R}\mathcal{D}_{G}^{L}=f_{\mu,\lambda}\cdot(\frac{1}{L_{1}^{L}},1,-\frac{1}{L_{2}^{L}},-1)+O(\epsilon^{2})$. For this calculation, we need to know $\mathcal{D}_{G}^{L}$ and $\mathcal{D}_{G}^{R}$ explicitly and it turns out that due to their special structure, multiplying them does change the vectors $\mathbf{v}^{L}$ and $dG_{1}^{R}$ much. We refer readers to equations (9.5), (9.6) and (9.10) of [GHX] for details of this calculation.

When taking into account the conservation of angular momentum $G_{0}+G_{1}+G_{2}=0$, in the reduced system of coordinates $(G_{0},\mathsf{g}_{01},G_{2},\mathsf{g}_{12})$. The details of this reduction are given in Section 9.3 of [GHX], which gives the derivatives of the global map in the isosceles limit $Y\to 0$

[TABLE]

where $\bar{\mathbf{u}}=(0,\frac{1}{\bar{L}_{2}^{L}},1,-\frac{1}{\bar{L}_{1}^{L}}-\frac{1}{\bar{L}_{2}^{L}})+O(\epsilon^{2}+\beta)$ and $\bar{\mathbf{l}}=(0,0,\frac{1}{L_{1}^{L}},1)+O(\epsilon^{2}+\beta)$. This gives the $O(\chi^{2})$ terms in the form of $d\mathbb{G}$ in Proposition 5.10.

C.2. The non-isosceles case

We next include the perturbations coming from the non-isoscelesness and complete the proof of Proposition 5.10. We shall control $|Y|\leq\nu$ for some small $\nu$, and show that the non-isoscelesness only contribute the $O(\nu f_{\mu,\lambda}\chi)$ error term to $d\mathbb{G}$. The main difficulty in the section is to prove (5.2) and (5.3), which needs a careful analysis of the special structure of the fundamental solution to the variational equation.

Proof of Proposition 5.10.

Between the section $\mathcal{S}_{+}$ and $\mathcal{S}^{M}$, let us consider the non-isosceles escaping orbits such that

[TABLE]

From (C.2) we know that the above conditions are satisfied if they are valid on the section $\mathcal{S}_{+}$. We compute the variational equation of the variables $(X,Y)$ where $X=(L_{0},\ell_{0},L_{2},\ell_{2})$ and $Y=(G_{0},g_{0},G_{1},g_{1},G_{2},g_{2})$ along such an orbit. Recall that we reduce the variable $L_{1}$ from the conservation of energy and treat the variable $\ell_{1}$ as the new time, we then have in the leading term

[TABLE]

where $K_{11}$ is the coefficient matrix (B.5) of the variational equation for $X$, and $K_{22}$ is the coefficient matrix in (C.1). We obtain the following estimates for its fundamental solution,

[TABLE]

where $\mathcal{D}_{2}^{L}$ is in (B.6), $\mathcal{D}_{G}^{L}$ is in (C.2), and $\mathcal{K}^{L}_{12}=\nu\epsilon^{2}\chi\bar{\mathbf{u}}_{1}\otimes\hat{\mathbf{l}}_{1}+O(\nu(\epsilon^{3}+\epsilon^{2}\beta)\chi)$ with $\bar{\mathbf{u}}_{1}$, $\bar{\mathbf{l}}_{1}$ from Proposition 3.10, $\hat{\mathbf{l}}_{1}=\bar{\mathbf{l}}_{1}\int_{0}^{\chi}K_{12}(s)\mathbb{Y}(s)\,ds$ and $\mathbb{Y}(s)$ being the fundamental solution of (C.1). We refer readers to Step 1 of Section 9.4 of [GHX] for details of this estimate.

For the immediate piece of the orbits, which leave the section $\mathcal{S}^{M}$ and then return to it, we have the the following estimate for the fundamental solution of the corresponding variational equation,

[TABLE]

where $\mathcal{D}^{I}$ is as in (B.9) and $\mathcal{D}_{G}^{I}$ is as in (C.3).

For the returning orbits between the section $\mathcal{S}^{M}$ and $\mathcal{S}_{-}$ satisfying the condition (C.6), we have the following estimate for the fundamental solution of the corresponding variational equation,

[TABLE]

where $\mathcal{D}_{R}$ is in (B.7), $\mathcal{D}_{G}^{R}$ is in (C.4), and $\mathcal{K}_{21}^{R}=O(\nu\epsilon^{2}f_{\mu,\lambda}\chi).$ We refer readers to Step 2 of Section 9.4 of [GHX] for details of this calculation, with modification being integrating $\ell_{1}$ over time interval $f^{-2}_{\mu,\lambda}\epsilon^{-1}$ to $f^{-2}_{\mu,\lambda}\chi$, for the same reason as the way we derive (C.4).

We next consider the transition from the left to right on the section $\mathcal{S}^{M}$. We get by equation (9.21) of [GHX]

[TABLE]

where $\mathcal{M}$ is the transition matrix for the $X^{L,R}$ given (B.12) and $\mathbf{v}^{R}\otimes dG_{1}^{R}$ is in Lemma C.1. The same structure holds true the right-left transition with obvious modification.

To give the structure of $d\mathbb{G}$ in the statement, we multiply the matrices obtained above. The nonisosceles $|Y|<\nu$ gives only $O(\nu\chi)$ contribution to $d\mathbb{G}$ and does not spoil the $O(\chi^{2})$ and $O(\chi)$ leading terms obtained in the isosceles limit. For (5.2) and (5.3) in the statement, we refer readers to Step 4 of Section 9.4 of [GHX] for the argument, which utilizes the special structure of the matrices (C.8) and (C.10). This completes the proof of the proposition. ∎

Appendix D Numeric verification of the transversality conditions

Since the dynamics near triple collision is nonperturbative, in particular, it is not easy to explicitly find $\gamma_{I}$ and $\gamma_{O}$ and integrate the variational equations along them, we choose to verify the transversality condition using a computer, as people have done in the literature (see [SM] etc). We only consider the case with 3 equal masses, that is, $m_{1}=m_{3}=m_{4}=1$. So in this case $m_{2}=3+o(1)$ for being admissible as in Definition 3.7. Then the equations of motion and the variational equations are concrete ODEs. The numeric task is to solve for $\gamma_{I}$ and $\gamma_{O}$, then integrate along them the corresponding variational equation with the stable and unstable directions at the fixed point as initial conditions. Explicit formulas for the variational equations can be find in [GHX, Section 8 and Appendix D].

**Step 1: ** We numerically compute111We used the ODE solver ’ode113’ in Matlab with ’RelTol’=$10^{-13}$ and ’AbsTol’=$10^{-15}$, and with the initial condition $O_{1}+10^{-14}\mathbf{e}^{u}$, where $\mathbf{e}^{u}$ is the unstable direction of the fixed point $O_{1}$ on the collision manifold. $\gamma_{O}$ (going to the side $\psi=\frac{\pi}{2}$) by solving I3BP on the collision manifold from a very small neighborhood of a fixed point along the unstable direction to the section $\mathcal{S}_{+}$, and integrate222We used the ODE solver ’ode113’ in Matlab with ’RelTol’=$10^{-7}$ and ’AbsTol’=$10^{-8}$. the corresponding variational equation along this orbit for the variables $(dw_{0},d\mathsf{g}_{01},dw_{2},d\mathsf{g}_{12})$ with initial conditions $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$, which are the unit eigenvectors corresponding to the two largest eigenvalues $\mu_{u}>0$ and $-\frac{v_{*}}{2}$, starting from a neighborhood of $O$. We perform the renormalization on the section $\mathcal{S}_{+}$ and then change from the coordinate $(dw_{0},d\mathsf{g}_{01},dw_{2},\mathsf{g}_{12})$ to $(dG_{0},d\mathsf{g}_{01},dG_{2},d\mathsf{g}_{12})$. Numerically, we get $E^{u}_{4}(\gamma(0))=\text{span}\{\hat{\mathbf{e}}_{1},\hat{\mathbf{e}}_{2}\}$, where $\hat{\mathbf{e}}_{1}=(0.5847,0.5608,0,0.5862)$ and $\hat{\mathbf{e}}_{2}=(0.4774,0.5282,-0.0701,0.6987)$.

Step 2: Next, we find the orbit $\gamma_{I}$ numerically. Note that for the I3BP, at the Lagrange fixed point $O$ with $v_{*}<0$, there are two linearly independent stable directions. One is on the collision manifold, which we denote as $\mathbf{e}_{1}^{s}$, the other is $\mathbf{e}_{2}^{s}=(v_{*},E,0,0)$, where $E$ is the total energy333By Proposition 3.8, now the total energy for the I3BP is $-1+o(1)$.. So we solve444We use the ODE solver ‘ode113’ in Matlab with ’RelTol’=$10^{-13}$ and ’AbsTol’=$10^{-14}$. I3BP backward along the stable direction with initial conditions of the form

[TABLE]

From the numeric result, there exist a consecutive set of initial conditions ($k=496,\dots,504$) for which the corresponding backward solutions satisfy the energy condition for $\gamma_{I}$ ($E_{1}$ is positive and of order $O(10^{-3})$) when the variable $r$ is of order $10^{3}$ and go to the side $\psi=\frac{\pi}{2}$. We then evaluate the corresponding variational equation backward along the corresponding to all these orbits with initial condition $E_{0}^{s}(O)$ and then change from the coordinate $(dw_{0},d\mathsf{g}_{01},dw_{2},\mathsf{g}_{12})$ to $(dG_{0},d\mathsf{g}_{01},dG_{2},d\mathsf{g}_{12})$ on the section $\mathcal{S}^{R}_{-}$. Numerically, we get that $E_{4}^{s}(\gamma_{I}(0))$ is spanned by $\{\hat{\mathbf{e}}_{3},\hat{\mathbf{e}}_{4}\}$, where $\hat{\mathbf{e}}_{3}=(0,0,0,1)$ and $\hat{\mathbf{e}}_{4}=(-0.7864,0.4473,0,0.4261)$. We also have $E^{s}_{3}(\gamma_{I}(0))=\text{span}\{\bar{\mathbf{e}}_{4}:=(-0.8692,0.4944)\}$

Step 3: *Checking the cone conditions in Proposition 5.10. * Since $\bar{\mathbf{l}}_{2}=(0,0,\frac{1}{L_{1}^{L}},1)+O(\beta)$, from the definition of $\bar{\mathbf{u}}_{3}$ and the numerics in Step 1, we have $\mathbf{u}=\frac{0.0701-0.6607L_{1}^{L}}{0.5862L_{1}^{L}}\hat{\mathbf{e}}_{1}+\hat{\mathbf{e}}_{2}+O(\beta)$ and setting the last two entries of $\mathbf{u}$ to zero we obtain $\bar{\mathbf{u}}_{3}$. Since by Proposition 3.6 $L_{1}^{L}=2/\sqrt{3}+o(1)$, we know that $\text{span}\{\bar{\mathbf{u}}_{3}\}\cap\mathcal{C}_{\frac{1}{100}}(E^{s}_{3}(\gamma_{I}(0)))=\{0\}$. Since $\bar{\mathbf{u}}_{2}=(0,0,-1,-\frac{1}{\bar{L}_{2}^{L}})+o(1)$ and by Proposition 3.8 now $\frac{1}{\bar{L}_{2}^{L}}=2/(9\sqrt{3})+o(1)$, with the numeric results from Step 2, we clearly have $\text{span}\{\bar{\mathbf{u}}_{2},\bar{\mathbf{u}}_{3}\}\cap\mathcal{C}_{\frac{1}{100}}(E^{s}_{4}(\gamma_{I}(0)))=\{0\}$.

Step 4: *Verifying Proposition 5.13. * We just keep track of $dw_{0}$ components of $E^{u}_{0}(\gamma_{O}(t))$, $t<0$ and $E^{s}_{0}(\gamma_{I}(t))$, $t>0$. The evolution of $dw_{0}$ and $\psi$ is illustrated in Figure 4. The values are close to nonzero constants when the orbits are in a neighborhood of the Lagrange fixed point as well as close to the sections $\mathcal{S}_{\pm}$, so they can only become zero at finitely many points. On the other hand, along $\gamma_{I}$ and $\gamma_{O}$, the variable $\psi$ also becomes zero at finitely many points, in particular it does not vanish in a neighborhood of the Lagrange fixed point. This reduces the problem to checking the non-coincidence of the zeros of $\psi\pm\pi/2$ and $w_{0}$ on two compact time intervals. Hence we have numerically justified the non-degeneracy in Proposition 5.13.