Estimates of products of norms for flows away from homoclinic tangency

TL;DR

This paper provides a detailed proof of estimates for products of norms in dynamical flows that are away from homoclinic tangency, contributing to the understanding of their stability properties.

Contribution

It offers a rigorous proof of norm product estimates for flows away from homoclinic tangency, albeit weaker than the expected bounds.

Findings

Established detailed estimates for products of norms in specific flows

Demonstrated the estimates are weaker than anticipated by experts

Contributed to the theoretical understanding of flow stability

Abstract

We give a detailed proof for the estimates of products of norms for flows away from homoclinic tangency. The estimates we can prove is weaker than the expectation of experts.

Equations

$$\Vert {\psi }_t|V^s(x)\Vert \Vert {\psi }_{-t}|V^c({\varphi }_t(x))+V^u({\varphi }_t(x))\Vert \le N{e}^{-\lambda t},$$

$$\Vert {\psi }_t|V^s(x)+V^c(x)\Vert \Vert {\psi }_{-t}|V^u({\varphi }_t(x))\Vert \le N{e}^{-\lambda t}.$$

$$\Vert {\psi }_{\tau }|V^s(x)\Vert \le N(1+\delta {)}^{-\tau },$$

$$\Vert {\psi }_{-\tau }|V^u(x)\Vert \le N(1+\delta {)}^{-\tau }.$$

$$0=t_0<t_1<\cdots <t_{\ell }=\tau$$

$$\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_{i+1}-t_i}|V^s({\varphi }_{t_i})\Vert \le N{\lambda }_1^{\tau },$$

$$\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_i-t_{i+1}}|V^s({\varphi }_{t_{i+1}})\Vert \le N{\lambda }_1^{\tau },$$

$$\Vert A\Vert \le n^{\frac{1}{2}}\Vert A{e}_i\Vert \le n|a_{ji}|$$

$$\Vert A\circ Q(v)\Vert \ge |(A\circ Q(v){)}_j|=|(A(v){)}_j+ϵt{a}_{ji}{v}_1|.$$

$$\Vert A\circ Q(v)\Vert \ge ϵ|a_{ji}{v}_1|\ge ϵn^{-\frac{3}{2}}\Vert A\Vert .$$

$$\frac{1+\gamma }{\gamma }2{ϵ}_1<{ϵ}^{\prime }.$$

$$T^{\prime }>\frac{\frac{3}{2}\ln n-\ln {ϵ}_1}{\ln (1+\frac{1}{2}\delta )}.$$

$$0=t_0<t_1<\cdots <t_{\ell }=\tau ,$$

$$\Vert {\psi }_{t_2-t_1}\circ Q_1(v_1)\Vert \ge {ϵ}_1{n}^{-\frac{3}{2}}\Vert {\psi }_{t_2-t_1}|V_1^s\Vert \Vert {\psi }_{t_1}|V^s(x)\Vert .$$

$$\Vert {\psi }_{t_3-t_2}\circ Q_2\circ {\psi }_{t_2-t_1}\circ Q_1\circ {\psi }_{t_1}|V^s(x)\Vert$$

$$\ge ({ϵ}_1{n}^{-\frac{3}{2}}{)}^2\Vert {\psi }_{t_3-t_2}|V_1^s\Vert \Vert {\psi }_{t_2-t_1}|V_1^s\Vert \Vert {\psi }_{t_1}|V^s(x).$$

$$\cdots \cdots$$

$$\Vert (\prod _{i=1}^{\ell -1}{\psi }_{t_{i+1}-t_i}|V_i^s\circ Q_i)\circ {\psi }_{t_1}|V^s(x)\Vert$$

$$\ge ({ϵ}_1{n}^{-\frac{3}{2}}{)}^{\ell -1}\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_{i+1}-t_i}|V_i^s\Vert$$

$$\ge ({ϵ}_1{n}^{-\frac{3}{2}}{)}^{\frac{\tau }{T^{\prime }}}\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_{i+1}-t_i}|V_i^s\Vert$$

$$\Vert L_1|V^s(x)\Vert \le N(1+\delta {)}^{-\tau }$$

$$\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_{i+1}-t_i}|V_i^s\Vert \le N[(1+\delta {)}^{-1}({ϵ}_1{n}^{-\frac{3}{2}}){]}^{-\frac{1}{T^{\prime }}}<N(\frac{1+\frac{1}{2}\delta }{1+\delta }{)}^{\tau }$$

$$\Vert (\prod _{i=1}^{\ell -1}{\psi }_{t_{i+1}-t_i}|V_i^s\circ Q_i)\circ {\psi }_{t_1}|V^s(x)\Vert$$

$$\ge ({ϵ}_1{n}^{-\frac{3}{2}}{)}^{\frac{\tau }{T^{\prime }}}\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_{i+1}-t_i}|V_i^s\Vert$$

$$\Vert {\psi }_{-t_1}\circ P_1\circ \cdots \circ {\psi }_{t_{\ell -2}-t_{\ell -1}}\circ P_{\ell -1}\circ {\psi }_{t_{\ell -1}-t_{\ell }}|V^u(x)\Vert$$

$$\ge ({ϵ}_1{n}^{-\frac{3}{2}}{)}^{\frac{\tau }{T^{\prime }}}\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_i-t_{i+1}}|V_{i+1}^u\Vert ,$$

$$\Vert P_1^{-1}\circ Q_i-\mathrm{id}\Vert \le \frac{1+\gamma }{\gamma }2{ϵ}_1<{ϵ}^{\prime }.$$

$$({ϵ}_1{n}^{-\frac{3}{2}}{)}^{\frac{\tau }{T^{\prime }}}\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_{i+1}-t_i}|V_i^s\Vert \le N(1+\delta {)}^{-\tau },$$

$$({ϵ}_1{n}^{-\frac{3}{2}}{)}^{\frac{\tau }{T^{\prime }}}\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_i-t_{i+1}}|V_{i+1}^u\Vert \le N(1+\delta {)}^{-\tau }.$$

$$\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_{i+1}-t_i}|V_i^s\Vert \le N{\lambda }_1^{\tau },$$

$$\prod _{i=0}^{\ell -1}\Vert {\psi }_{t_i-t_{i+1}}|V_{i+1}^u\Vert \le N{\lambda }_1^{\tau }.$$

Full text

Qianying Xiao

1 Introduction

Assume $X\in\mathcal{X}^{1}(M)$ is away from homoclinic tangencies. $\phi_{t}$ is the flow generated by $X$, $\psi_{t}$ is the linear Poincaré flow. There exists a neighborhood $\mathcal{U}$ of $X$ such that any $Y\in\mathcal{U}$ has no homoclinic tangencies.

Wen [2] shows that there exist $\mathcal{U}_{2}\subset\mathcal{U}$, $\gamma>0$, $N>0$, $\lambda>0$, $\delta>0$, and $T>0$, any $Y\in\mathcal{U}$, any periodic orbits $orb(x)$ of $Y$ with period $\tau>T$,

$\psi_{\tau}(x)$ has at most one eigenvalue with modulo in $[(1+\delta)^{-\tau},(1+\delta)^{\tau}]$, i.e. there exists eigenspace splitting $\mathcal{N}_{x}=V^{s}(x)+V^{c}(x)+V^{u}(x)$ such that $V^{c}(x)$ is the eigenspace corresponding to the eigenvalue with modulo in $[(1+\delta)^{-\tau},(1+\delta)^{\tau}]$, and $\dim V^{c}(x)\leq 1$. 2. 2.

$\angle(V^{s}(x),V^{c}(x)+V^{u}(x))>\gamma$, $\angle(V^{s}(x)+V^{c}(x),V^{u}(x))>\gamma$. 3. 3.

The splitting is dominated:

[TABLE]

[TABLE] 4. 4.

Estimation of norms of products:

[TABLE]

[TABLE]

The above estimates are improved by Wen [3]. In  [3, Lemma 3.4], Wen gives the estimates of products of norms, which are stronger than the previous result. Wen regards the proofs are standard in Liao’s and Mane’s work( for instance see [1, Page 528]), and he doesn’t give details of the proofs there. We try to prove it as an exercise, and it turns out we are not able to prove the Item 2 of  [3, Lemma 3.4]. Instead we prove a weaker version, and it works for our purpose to prove Theorem 2.1 of our work  [4]. The following ideas of Liao: minimal nonhyperbolic set, (local) star flows and the Selecting lemma, which are stressed by Wen extensively, help us to avoid a direct use of the non-proved estimates.

Proposition 1**.**

There exists neighborhood $\mathcal{U}_{3}\subset\mathcal{U}_{2}$ of $X$, $0<\lambda_{1}<1$, $T^{\prime}>0$, any $Y\in\mathcal{U}_{3}$, any periodic point $x$ of $Y$ with period $\tau>\max\{T,T^{\prime}\}$, any partition of $[0,\tau]$:

[TABLE]

such that $t_{i+1}-t_{i}\geq T^{\prime}$ for $0\leq i\leq\ell-1$, the following are satisfied: either

[TABLE]

or

[TABLE]

2 The proofs

Let us give the details of estimating the products of norms. For the sake of completeness, let us insert a well-known fact in linear algebra. Let $E^{n}$ denotes the $n$-dimensional Euclidean space. $A:E^{n}\rightarrow E^{n}$ is a linear map.

Lemma 2.1**.**

[1, Page 528]** Any $\epsilon>0$, $v\in E^{n}$ with $\|v\|=1$, there exist $Q:E^{n}\rightarrow E^{n}$ such that $\|Q-\mathrm{id}\|\leq\epsilon$, and that $\|A\circ Q(v)\|\geq\epsilon n^{-\frac{3}{2}}\|A\|$.

Proof.

Let $\{e_{1},\cdots,e_{n}\}$ be an orthonormal basis, $v=(v_{1},\cdots,v_{n})$. We may as well assume $|v_{1}|\geq n^{-\frac{1}{2}}\|v\|$.

$a_{ji}=(Ae_{i})_{j}$ such that

[TABLE]

.

$Q:E^{n}\rightarrow E^{n}$ such that $Q(e_{1})=e_{1}+\epsilon te_{i}$, $t=1$ or $-1$, and $Q(e_{k})=e_{k}$ for $k>1$. Obviously $\|Q-\mathrm{id}\|\leq\epsilon$.

[TABLE]

Choose $t$ such that $(A(v))_{j}\cdot\epsilon ta_{ji}v_{1}\geq 0$, then

[TABLE]

∎

Proof of the Proposition.

Let $\epsilon>0$, $\mathcal{U}_{3}\subset\mathcal{U}_{2}$ be determined by Frank’s lemma. $K$ is an upperbound of $\|\psi^{Y}_{t}\|$ for $Y\in\mathcal{U}_{3}$, $1\leq t\leq 2$.

$\epsilon^{\prime}=\epsilon K^{-1}$, $\epsilon_{1}>0$ such that

[TABLE]

[TABLE]

Given any $Y\in\mathcal{U}_{3}$, any periodic point $x$ of $Y$ with periodic $\tau\geq\max\{T^{\prime},T\}$, any partition of $[0,\tau]$:

[TABLE]

such that $t_{i+1}-t_{1}\geq T^{\prime}$ for $0\leq i\leq\ell-1$.

• Case 1

Assume $\dim V^{c}(x)=1$. Denote $\mathcal{N}_{\phi_{t_{i}}(x)}$ as $N_{i}$, $V^{\sigma}_{\phi_{t_{i}}(x)}$ as $V^{\sigma}_{i}$ for $\sigma=s,c,u$, $i=0,\cdots,\ell-1$.

Define $Q_{1}:N_{1}\rightarrow N_{1}$ such that $Q_{1}|V^{c}_{1}+V^{u}_{1}=\mathrm{id}$, $Q_{1}(V^{s}_{1})=V^{s}_{1}$, $\|Q_{1}|V^{s}_{1}-\mathrm{id}\|<\epsilon_{1}$, $v_{1}\in V^{s}_{1}$ such that $\|v_{1}\|=\|\psi_{t_{1}}|V^{s}(x)\|$, and

[TABLE]

Moreover, $\|Q_{1}-\mathrm{id}\|<\frac{1+\gamma}{\gamma}\epsilon_{1}<\epsilon^{\prime}$.

Similarly, we can define $Q_{2}:N_{2}\rightarrow N_{2}$ such that $\|Q_{2}-\mathrm{id}\|<\epsilon^{\prime}$, $Q_{2}|V^{c}_{2}+V^{u}_{2}=\mathrm{id}$, $Q_{2}(V^{s}_{2})=V^{s}_{2}$, and that

[TABLE]

[TABLE]

[TABLE]

$Q_{\ell-1}:N_{\ell-1}\rightarrow N_{\ell-1}$, such that $\|Q_{\ell-1}-\mathrm{id}\|<\epsilon^{\prime}$, $Q_{\ell-1}|V^{c}_{\ell-1}+V^{u}_{\ell-1}=\mathrm{id}$, $Q_{\ell-1}(V^{s}_{\ell-1})=V^{s}_{\ell-1}$.

[TABLE]

Denote $L_{i,r}=\psi_{t_{i+1}-t_{i}}\circ(\mathrm{id}+r(Q_{i}-\mathrm{id}))$ for $i=1,\cdots,\ell-1$, $L_{r}=L_{\ell-1,r}\circ\cdots\circ L_{1,r}\circ L_{0,r}$ $(0\leq r\leq 1)$.

Claim**.**

The spectral radius of $L_{1}|V^{s}(x)$ is less than $(1+\delta)^{-\tau}$. Otherwise, for some $r$, $L_{r}$ admits at least two eigenvalues with modulo in $[(1+\delta)^{-\tau},(1+\delta)^{\tau}]$. By Frank’s lemma, there exists $Z\in\mathcal{U}_{2}$ such that $\psi^{Z}_{\tau}(x)=L_{r}$, a contradiction.

Apply Frank’s lemma, there exist $Z^{\prime}\in\mathcal{U}_{2}$ such that $Z^{\prime}=Y$ along $orb(x)$, $\psi^{Z^{\prime}}_{\tau}(p)=L_{1}$, $V^{\sigma}_{Z^{\prime}}(x)=V^{\sigma}(x)$, $\sigma=s,c,u$. Hence

[TABLE]

According to inequality 1, and by $T^{\prime}>\frac{\frac{3}{2}\ln n-\ln\epsilon_{1}}{\ln(1+\frac{1}{2}\delta)}$,

[TABLE]

Let $\lambda_{1}=\frac{1+\frac{1}{2}\delta}{1+\delta}$, and we are done.

• Case 2

Assume $\dim V^{c}(x)=0$. As in case 1, we have $Q_{i}$ for $i=1,\cdots,\ell-1$, such that $Q_{i}|V^{u}_{i}=\mathrm{id}$, $Q_{i}(V^{s}_{i})=V^{s}_{i}$, and $\|Q_{i}|V^{s}_{i}-\mathrm{id}\|<\epsilon_{1}$. Moreover,

[TABLE]

[TABLE]

Similarly, we have $P_{i}$ for $i=1,\cdots,\ell-1$, such that $P_{i}|V^{s}_{i}=\mathrm{id}$, $P_{i}(V^{u}_{i})=V^{u}_{i}$, and $\|P_{i}|V^{u}_{i}-\mathrm{id}\|<\epsilon_{1}$, and

[TABLE]

[TABLE]

[TABLE]

For $0\leq\alpha\leq 1$, $0\leq\beta\leq 1$, define $L_{i,\alpha,\beta}=\psi_{t_{i+1}-t_{i}}\circ(\mathrm{id}+\alpha(P^{-1}_{i}-\mathrm{id}))\circ(\mathrm{id}+\beta(Q_{i}-\mathrm{id})).$

Denote $L_{\alpha,\beta}=L_{\ell-1,\alpha,\beta}\circ\cdots\circ L_{0,\alpha,\beta}$.

Claim**.**

Either the spectral radius of $L_{1,1}|V^{s}(x)$ is less than $(1+\delta)^{-\tau}$, or the spectral radius of $L_{1,1}|V^{u}(x)$ is greater than $(1+\delta)^{\tau}$. Otherwise some $L_{\alpha,\beta}$ admits two eigenvalues (counting multiplicity) with modulo in $[(1+\delta)^{-\tau},(1+\delta)^{\tau}]$. Apply Frank’s lemma, there exists $Z\in\mathcal{U}_{2}$ such that $\psi^{Z}_{\tau}(x)$ has two eigenvalues with modulo in $[(1+\delta)^{-\tau},(1+\delta)^{\tau}]$, a contradiction.

By Frank’s lemma, there exists $Z^{\prime}\in\mathcal{U}_{2}$, such that $\psi^{Z^{\prime}}_{\tau}(x)=L_{1,1}$, therefore $\|L_{1,1}|V^{s}(x)\|\leq N(1+\delta)^{-\tau}$ or $\|L_{1,1}^{-1}|V^{u}(x)\|\leq N(1+\delta)^{-\tau}$.

Note that $L_{1,1}|V^{s}(x)=(\prod^{\ell-1}_{i=0}\psi_{t_{i+1}-t_{i}}\circ Q_{i})|V^{s}(x)$, $L_{1,1}^{-1}|V^{u}(x)=\psi_{-t_{1}}\circ\prod^{1}_{i=\ell-1}P_{i}\circ\psi_{t_{i}-t_{i+1}}|V^{u}(x)$.

Hence

[TABLE]

or

[TABLE]

Let $\lambda_{1}=\frac{1+\frac{1}{2}\delta}{1+\delta}$, then

[TABLE]

or

[TABLE]

∎