On volume preserving almost Anosov flows

TL;DR

This paper establishes limit laws for volume-preserving almost Anosov flows on 3-manifolds with neutral cubic saddle periodic points, including estimates for Dulac maps in planar vector fields.

Contribution

It introduces new limit laws for a specific class of flows and derives estimates for Dulac maps related to cubic neutral saddles.

Findings

Limit laws for volume-preserving almost Anosov flows on 3-manifolds.

Estimates for Dulac maps for cubic neutral saddles.

Analysis of the dynamics near neutral periodic points.

Abstract

The purpose of this paper is to establish limit laws for volume preserving almost Anosov flows on $3$-three manifolds having a neutral periodic of cubic saddle type. In the process, we derive estimates for the Dulac maps for cubic neutral saddles in planar vector fields.

Equations

$$\left(\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\end{array}\right)=X\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}x(a_0{x}^2+a_{1}xy+a_2{y}^2))\\ -y(b_0{x}^2+b_{1}xy+b_2{y}^2))\\ 1+w(x,y)\end{array}\right)+\mathcal{O}(4)$$

$$a_1,b_1\in \mathbb{R},a_0,a_2,b_0,b_2\ge 0\text{ with }\Delta :=a_2{b}_0-a_0{b}_2\ne 0\text{ and }{c}_1^2<4c_0{c}_2$$

$$\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)=X_{hor}\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}x(a_0{x}^2+a_{1}xy+a_2{y}^2)\\ -y(b_0{x}^2+b_{1}xy+b_2{y}^2)\end{array}\right)+\mathcal{O}(4),$$

$$\frac{b_1}{a_1}=\frac{b_2{a}_0+a_2+2b_0{b}_2}{b_2{a}_0+a_2+2a_0{a}_2}.$$

$$3a_0=b_0,a_2=3b_2,a_1=b_1.$$

$${\beta }_0:=\frac{a_0+b_0}{2a_0},{\beta }_2:=\frac{a_2+b_2}{2b_2},{\beta }_{\ast }=\frac{1}{2}\min \left\{1,\frac{a_2}{b_2},\frac{b_0}{a_0}\right\}.$$

$$\xi (\eta ,\tilde{T})={\xi }_0(\eta ){\tilde{T}}^{-{\beta }_2}(1+O({\tilde{T}}^{-{\beta }_{\ast }},T^{-\frac{1}{2}}\log T)).$$

$$\omega (\eta ,\tilde{T})={\omega }_0(\eta ){\tilde{T}}^{-{\beta }_0}(1+O({\tilde{T}}^{-{\beta }_{\ast }},T^{-\frac{1}{2}}\log T)).$$

$$\omega =D(\xi )={\omega }_0(\eta ){\xi }_0(\eta {)}^{-\frac{{\beta }_0}{{\beta }_2}}{\xi }^{\frac{{\beta }_0}{{\beta }_2}}\left(1+\mathcal{O}({\xi }^{\frac{{\beta }_{\ast }}{{\beta }_2}},-{\xi }^{\frac{1}{2{\beta }_2}}\log \xi )\right).$$

$$\left(\begin{array}{c}\dot{\bar{x}}\\ \dot{\bar{y}}\\ \dot{\bar{x}}\end{array}\right)=\left(\begin{array}{c}\bar{x}({\bar{x}}^2+\gamma \bar{x}\bar{y}+3{\bar{y}}^2))\\ -\bar{y}(3{\bar{x}}^2+\gamma \bar{x}\bar{y}+{\bar{y}}^2))\\ 1+\bar{w}(\bar{x},\bar{y})\end{array}\right)+\mathcal{O}(4).$$

$$\frac{{\int }_0^{t}v\circ {\varphi }^{s}ds-t\int vdVol}{\sqrt{t\log t}}{\Rightarrow }_{dist}\mathcal{N}(0,{\sigma }^2)\text{ as }t\to \infty ,$$

$$f\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}x(1+a_0{x}^2+a_{1}xy+a_2{y}^2)\\ y(1-b_0{x}^2-b_{1}xy-b_2{y}^2)\\ z+O(|w(x,y)|)\end{array}\right)+\mathcal{O}(3),$$

$$\phi (z)=\min \{n\ge 1:f_{hor}^n(z)\notin P_0\}$$

$$\{\begin{array}{l}\dot{x}=x(a_0{x}^2+a_{1}xy+a_2{y}^2),\\ \dot{y}=-y(b_0{x}^2+b_{1}xy+b_2{y}^2),\end{array}$$

$$\{\begin{array}{l}(u+2)a_0=v{b}_0\\ (v+2)b_2=u{a}_2\end{array}\text{ that is:}\{\begin{array}{l}u=\frac{2b_2{c}_0}{\Delta },\\ v=\frac{2a_0{c}_2}{\Delta }.\end{array}$$

$${\beta }_0:=\frac{a_0+b_0}{2a_0}=\frac{u+v+2}{2v},{\beta }_2:=\frac{a_2+b_2}{2b_2}=\frac{u+v+2}{2u},\frac{{\beta }_0}{{\beta }_2}=\frac{u}{v},$$

$$\xi (\eta ,T)={\xi }_0(\eta )T^{-{\beta }_2}\left(1-{\xi }_1(\eta )T^{-1}+O(T^{-2},T^{-2{\beta }_2})\right)$$

$$\omega (\eta ,T)={\omega }_0(\eta )T^{-{\beta }_0}\left(1-{\omega }_1(\eta )T^{-1}+O(T^{-2},T^{-2{\beta }_0})\right).$$

$$L(x,y)=\{\begin{array}{ll}{x}^u{y}^v(\frac{a_0}{v}{x}^2+\frac{a_1}{v+1}xy+\frac{b_2}{u}{y}^2)& \text{ if }\Delta >0,\\ x^{-u}{y}^{-v}(\frac{a_0}{v}{x}^2+\frac{a_1}{v+1}xy\frac{b_2}{u}{y}^2{)}^{-1}& \text{ if }\Delta <0,\end{array}$$

$$L(x,y)=x^u{y}^v(\frac{b_0}{u+2}{x}^2+\frac{b_2}{u}{y}^2)=x^u{y}^v(\frac{a_0}{v}{x}^2+\frac{a_2}{v+2}{y}^2).$$

$$\dot{L}$$

$$\dot{M}=-M(c_0+c_{1}M+c_2{M}^2)x^2.$$

$${\xi }^u{\eta }^v(\frac{a_0}{v}{\xi }^2+\frac{a_1}{v+1}\xi \eta +\frac{b_2}{u}{\eta }^2)$$

$$\{\begin{array}{l}\frac{a_0}{v}+\frac{a_1}{v+1}M+\frac{b_2}{u}{M}^2=\frac{\Delta }{2c_0{c}_2}(c_0+c_{1}M+c_2{M}^2),\\ \frac{a_0{\xi }^2}{v}+\frac{a_1}{v+1}\xi \eta +\frac{b_2{\eta }^2}{u}=\frac{\Delta }{2c_0{c}_2}(c_0{\xi }^2+c_1\xi \eta +c_2{\eta }^2).\end{array}$$

$$x^2$$

$$\dot{M}=-G{M}^{1-\frac{1}{{\beta }_0}}{\left(c_0+c_{1}M+c_2{M}^2\right)}^{\frac{1}{2{\beta }_0}+\frac{1}{2{\beta }_2}}$$

$$G=G(\xi ,\eta ):={\xi }^{\frac{1}{{\beta }_2}}{\eta }^{\frac{1}{{\beta }_0}}{\left(c_0{\xi }^2+c_1\xi \eta +c_2{\eta }^2\right)}^{1-\frac{1}{2{\beta }_0}-\frac{1}{2{\beta }_2}}.$$

$${\int }_{\omega (\eta ,T)/{\zeta }_0}^{\eta /\xi (\eta ,T)}\frac{M^{\frac{1}{{\beta }_0}-1}dM}{{\left(c_0+c_{1}M+c_2{M}^2\right)}^{\frac{1}{2{\beta }_0}+\frac{1}{2{\beta }_2}}}=G(\xi (\eta ,T),\eta )T.$$

$${\xi }^u{\eta }^v(c_0{\xi }^2+c_1\xi \eta +c_2{\eta }^2)={\zeta }_0^u{\omega }^v(c_0{\zeta }_0^2+c_1{\zeta }_0\omega +c_2{\omega }^2).$$

Full text

Mathematics Subject Classification (2010): 37C10, 37D20, 37D25, 60F05

Keywords. Dulac map, almost Anosov flows, limit laws, stable laws, Central Limit Theorem, non-uniform hyperbolicity

1 Introduction

A flow $\phi^{t}:{\mathcal{M}}\times{\mathbb{R}}\to{\mathcal{M}}$ on a (in our setting $3$-dimensional) compact differentiable manifold ${\mathcal{M}}$ is called Anosov if its tangent bundle has a continuous flow-invariant mutually transversal splitting into a neutral flow direction $E^{c}$, a hyperbolically stable direction $E^{s}$ and a hyperbolically unstable direction $E^{u}$. The uniform hyperbolicity of such flows enables one to show various ergodic and statistical properties, such as ergodicity (if the flow is topologically mixing) and the Central Limit Theorem (CLT) for Hölder continuous observables.

We obtain an almost Anosov flow (see Definition 1.1 below) by inserting a neutral orbit $\Gamma\simeq\{(0,0)\}\times{\mathbb{S}}^{1}$ near which the flow has the following form in local Euclidean coordinates:

[TABLE]

where ${\mathcal{O}}(4)$ indicates terms of order four and higher, and the parameters satisfy

[TABLE]

for $c_{i}:=a_{i}+b_{i},\ i=0,1,2$. That is, the vector field is cubic in the transversal direction to $\Gamma$, but this is the only source of non-hyperbolicity. Finally, $w$ is a linear combination of homogeneous functions in $x$ and $y$, vanishing at $(0,0)$. Thus period of $\Gamma$ is its length.

The original motivation to study such system was to have a class of natural examples of non-uniformly hyperbolic invertible maps (think of the Poincaré map on a section $\Sigma\subset{\mathbb{R}}^{2}\times\{0\}$ or the time-$1$ map $f_{hor}=\phi_{hor}^{1}$ for the horizontal flow where only the $x$ and $y$ coordinates are taken into account:

[TABLE]

with the restrictions (2), as natural examples where operator renewal theory can be applied to get precise statistical laws for the flow. Initially, in [2] for the parameter range $\beta_{2}:=\frac{a_{2}+b_{2}}{2b_{2}}\leq 1$ where $f_{hor}$ preserves an infinite Sinai-Bowen-Ruelle (SRB) measure, we gave mixing rates for $C^{1}$ observables. Later [3], and more relevant to this paper, in the parameter range $\beta_{2}>1$ where the flow $\phi^{t}$ preserves a finite SRB-measure, we established limit laws (Stable Laws and the CLT with standard or non-standard scaling, depending on whether $\beta_{2}\in(1,2)$, $\beta=2$ or $\beta_{2}>2$).

All these results were obtained in the absence of mixed terms, i.e., $a_{1}=b_{1}=0$ in (1). This is of course not a natural assumption, and to our knowledge there is no change of coordinates that allows one to remove the mixed terms. In fact, if $c_{1}^{2}>4c_{1}c_{2}$, then the behaviour near the saddle is locally non-conjugate to the behaviour when $c_{1}^{2}<4c_{1}c_{2}$.

The purpose of this paper is to perform the analysis when mixed terms are present. The crux of the analysis is the existence of a local first integral (and its explicit form when ${\mathcal{O}}(4)$-terms are absent in (9)), which allows us to reduce the ODE to dimension one. We will show in Lemma 2.1 that the first integral $L$ can be found if

[TABLE]

This is a co-dimension one condition in parameter space. However, if we also stipulate that the flow $\phi^{t}$ is volume preserving, we must assume that $\mbox{div}\,X=0$ in (1), which is equivalent to $\mbox{div}\,{\mathcal{O}}(4)=0$ together with

[TABLE]

From these conditions, (4) follows automatically, and therefore (1) describes a generic volume preserving almost Anosov flow with a single neutral periodic orbit of cubic saddle type. We present the results on limit laws in the volume preserving setting, see Corollary 1.1.

Central to the proof is the analysis of the Dulac map near the neutral equilibrium of (3). This means that we take an incoming and an outcoming transversal to the flow, in our case an unstable leaf $W^{u}(0,\eta)$, $\eta\in[\eta_{0},\eta_{1}]$, and a stable leaf $W^{s}(\zeta_{0},0)$, see Figure 1, and the Dulac map $D:W^{u}(0,\eta)\to W^{s}(\zeta_{0},0)$ assigns the first intersection $\phi_{hor}^{T}(\eta,\xi_{0})$ of the integral curve through $(\eta,\xi_{0})$ with the outgoing transversal $W^{s}(\zeta_{0},0)$, and the corresponding flow-time is denoted as $T$. The main technical result of this paper are precise estimates of the Dulac map when (3) contains mixed terms, but using the assumption (4).

Dulac [5] introduced his map as an ingredient to prove that polynomial vector fields in the plane have at most finitely many limit cycles, thus making a major contribution to the solution of Hilbert’s 16th problem. Écalle [6] and Il’yashenko [8] independently corrected some weak parts in Dulac’s arguments, see also the summary in Roussarie’s book [14, Chapter 3 and Section 3.3]. Hilbert’s problem reduces to Dulac’s problem, namely that polycycles (i.e., heteroclinic saddle connections) cannot accumulated upon by limit cycles, and a crucial use of Dumortier’s blow-up theorem [4] allows one to restrict the attention to hyperbolic saddles. More recent contributions in this direction are by Mardešić and collaborators [9, 10, 11, 12, 15].

Our estimates only concern a single neutral saddle, and although for the purpose of Dulac’s problem they can be treated by blow-ups, precise formulas for the Dulac times (and hence the Dulac map, see (6)), at cubic saddles in this generality seem to be new.

1.1 Main results

The crucial estimates here are of the Dulac times, i.e., the times that orbits take to pass from an “incoming” unstable transversal to an “outgoing” unstable transversal to the flow, see Figure 1.

Theorem 1.1

Consider a $C^{3}$ vector field of local form (3) with parameters satisfying (2) and (4). Define

[TABLE]

Then there constants111The precise values of $\xi_{0}(\eta)$ and $\omega_{0}(\eta)$ are given in in the proof Proposition 2.1. $\xi_{0}(\eta),\omega_{0}(\eta)$ such that the following asymptotics hold:

[TABLE]

and

[TABLE]

as $T\to\infty$.

In particular, the functions $\xi$ and $\omega$ are regularly varying of order $\beta_{2}$ in $T$, that is $\lim_{T\to\infty}\frac{\xi(\eta,cT)}{\xi(\eta,T)}=c^{\beta_{2}}$ for every $c>0$ and analogous for $\omega(\eta,T)$. Moreover, the Dulac map $D:W^{u}(0,\eta)\to W^{s}(\zeta_{0},0)$ itself has the form (as $\xi\to 0$)

[TABLE]

With assumptions (5) and $c_{1}^{2}<4c_{0}c_{2}$ in place, we can use the change of coordinates $\bar{x}=\sqrt{a_{0}}x,\ \bar{y}=\sqrt{b_{2}}y$ and $\gamma=a_{1}/\sqrt{a_{0}b_{2}}\in(-4,4)$ to transform (1) into the one-parameter family

[TABLE]

for some transformed function $\bar{w}$.

Because of this genericity and reduced number of technicality that Lebesgue measure gives as opposed to SRB-measure, we state our statistical result for volume preserving flows. Theorem 1.1 is used to estimate the measures of the strips $\{\varphi=n\}$, see Figure 2, which in turn, together with the spectral properties of an induced Poincaré map $\hat{f}$ are crucial ingredients for the analysis required to establish the following stochastic limit properties of the flow $\phi^{t}$.

Corollary 1.1

Consider a volume preserving almost Anosov flow (7) on ${\mathcal{M}}$ with $\gamma\in(-4,4)$ and an observables $v:{\mathcal{M}}\to{\mathbb{R}}$ that is $C^{1}$ on ${\mathcal{M}}\setminus\Gamma$ and has the form $v=v_{0}+o(\rho)$ where $\int_{0}^{\tau}v_{0}\circ\phi^{t}\,dt$ is homogeneous of order $\rho\in(-2,1)$ in local coordinates $(x,y)$ near $p$ and $o(\rho)$ stands for terms of order $>\rho$.

If $\rho=0$, then $v$ satisfies the Central Limit Theorem with non-standard scaling $\sqrt{t\log t}$, i.e.,

[TABLE]

and the variance $\sigma^{2}>0$ unless $\int_{0}^{\tau}v\circ\phi^{t}\,dt$ is a coboundary. 2. 2.

If $\rho>0$, then $v$ satisfies the Gaussian Central Limit Theorem, i.e., with standard scaling $\sqrt{t}$. 3. 3.

If $\rho\in(-2,0)$ then $v$ satisfies a Stable Law of order $\frac{4}{2-\rho}\in(1,2)$.

Theorem 1.1 allows also to derive other limit theorems such as in the infinite measure setting of [2], but with mixed terms. But since we restrict to the Lebesgue measure (rather than SRB-measure) preserving case, we don’t give any further details.

1.2 Set-up

The set-up here is largely taken over from [3]. Our phase space will be the $3$-dimensional compact manifold ${\mathcal{M}}$.

Definition 1.1

[7*, Definition 1]**

A diffeomorphism $f:{{\mathbb{T}}}^{2}\to{{\mathbb{T}}}^{2}$ is called almost Anosov if there exists two continuous families of non-trivial cones $x\to{\mathcal{C}}_{x}^{u},{\mathcal{C}}_{x}^{s}$ such that except for a finite set $S$,*

• i)

$Df_{x}{\mathcal{C}}_{x}^{u}\subseteq{\mathcal{C}}_{f(x)}^{u}$* and $Df_{x}{\mathcal{C}}_{x}^{s}\supseteq{\mathcal{C}}_{f(x)}^{s}$;*

• ii)

$|Df_{x}v|>|v|$* for any $0\neq v\in{\mathcal{C}}_{x}^{u}$ and $|Df_{x}v|<|v|$ for any $0\neq v\in{\mathcal{C}}_{x}^{s}$.*

For $x\in S$, $Df_{x}$ is the identity.

A flow $f^{t}$ on $3$-torus ${\mathbb{T}}^{3}$ is called almost Anosov flow if it has a finite set $S$ of neutral periodic orbits, but everywhere else observes the condition of an Anosov flow in that there is a continuous splitting of the tangent bundle into a stable, an unstable and a neutral (flow) direction. For $x\in S$, the derivative at the return time $\tau$ is $Df^{\tau}_{x}$ is the identity.

The time-$1$ map $f$ of the flow $\phi^{t}$ of (1) has the form of a skew-product

[TABLE]

see [2, Section 2.1]. Restricted to the $(x,y)$-coordinates, this map $f_{hor}$ is a smooth almost Anosov map with a single neutral fixed point $p=(0,0)$. Let $\{P_{i}\}_{i=0}^{k}$ be the Markov partition for $f_{hor}$ (which we can assume to exist since $f_{hor}$ is a local perturbation of a Anosov diffeomorphism on ${\mathbb{T}}^{2}$). We assume that $p$ belongs to the interior of $P_{0}$. Clearly, the horizontal and vertical axes are the unstable and stable manifolds of $p$ respectively. We assume that the Markov partition element $P_{0}\subset U$ is a small rectangle such that $\overline{f_{hor}^{-1}(P_{0})\cup P_{0}\cup f_{hor}(P_{0})}\subset U$. Due to the symmetries $(x,y)\mapsto(\pm x,\pm y)$, it suffices to do the analysis only in the first quadrant $Q=[0,\zeta_{0}]\times[0,\eta_{0}]$ of $P_{0}$, see Figure 2. Without loss of generality (see [2, Lemma 2.1]) we can think of $[0,\zeta_{0}]\times\{\eta_{0}\}$ as a local unstable leaf and $\{\zeta_{0}\}\times[0,\eta_{0}]$ as a local stable leaf of the global diffeomorphism.

We consider an induced map $F_{hor}=f_{hor}^{\varphi}:Y\to Y$ for $Y:={{\mathbb{T}}}^{2}\setminus P_{0}$, where

[TABLE]

is the first return time to $Y$. Note that $F_{hor}$ is invertible because $f_{hor}$ is. In the first quadrant of $U\setminus P_{0}$, $\{\varphi=n\}:=\{z\in f^{-1}(Q)\setminus Q:\varphi(z)=n\}$, $n\geq 2$, are vertical strips adjacent to the local unstable leaf $[0,\zeta_{0}]\times\{\eta_{0}\}$, and converging to $\{0\}\times[\eta_{0},\eta_{1}]$ as $n\to\infty$. The images $F_{hor}(\{\varphi=n\})$ are horizontal strips, adjacent to the local stable leaf $\{\zeta_{0}\}\times[0,\eta_{0}]$, and converging to $[\zeta_{0},\zeta_{1}]\times\{0\}$ as $n\to\infty$, see Figure 2.

In contrast to $f_{hor}$, the induced map $F_{hor}$ is uniformly hyperbolic, but only piecewise continuous. Indeed, continuity fails at the boundaries of the strips $\{\varphi=n\}$, $n\geq 2$ (and $F$ is undefined on $W^{s}(p)$), but these boundaries are local stable and unstable leaves, and it is possible to create a countable Markov partition refining $\{P_{i}\}_{i=1}^{k}$ of $Y$ for $F$, in which all the strips $\{\varphi=n\}$ are partition elements.

2 Regular variation of $\mu(\varphi>n)$ with mixed terms

In this section, we allow quadratic mixed terms in (3), but for the moment leave out the ${\mathcal{O}}(4)$-terms. That is, we consider

[TABLE]

that is, (3) without the ${\mathcal{O}}(4)$ terms but with the restrictions (2) and (4). The condition $c_{1}^{2}<4c_{0}c_{2}$ avoids the formation of invariant lines $y=px$, but in the below proofs it is used to guarantee that expressions as $c_{0}+c_{1}M+c_{2}M^{2}$ for $M=y/x$ are positive. Our exposition closely follows [2], but since the mixed terms require slight adjustments throughout the proof, we will give it in full.

Let $u,v\in{\mathbb{R}}$ be the solutions of the linear equations

[TABLE]

Note that $u,v$ and $\Delta$ (recall $\Delta\neq 0$) all have the same sign and (4) implies that $\frac{b_{1}}{a_{1}}=\frac{u+1}{v+1}$. Compute that

[TABLE]

and note that $\beta_{0},\beta_{2}>\frac{1}{2}$ (or $=\frac{1}{2}$ if we allow $b_{0}=0$ or $a_{2}=0$ respectively). Under the extra assumption (5) we obtain $\beta_{0}=\beta_{2}=2$ and $u=v=1$.

The first estimates is about the Dulac map of (3).

Proposition 2.1

Consider a vector field on the $2$-torus with local form (3) for $a_{0},a_{2},b_{0},b_{2}\geq 0$ and $\Delta\neq 0$. There are functions $\xi_{0}(\eta),\omega_{0}(\eta),\xi_{1}(\eta),\omega_{1}(\eta)>0$ independent of $T$ (with exact expressions given in the proof) such that

[TABLE]

and

[TABLE]

Lemma 2.1

The function

[TABLE]

is a first integral of (3).

Proof of Lemma 2.1. First assume $\Delta>0$, so $u,v>0$ as well. By (10), we can write $L(x,y)$ as

[TABLE]

Using these two equivalent expressions and that $\frac{a_{1}}{v+1}=\frac{b_{1}}{u+1}$. by (4), we compute the Lie derivative directly

[TABLE]

Any function of a first integral is a first integral, in particular this holds for $1/L$. Therefore the conclusion is immediate for $\Delta<0$ too. $\Box$

Proof of Proposition 2.1. We carry out the proof for $\Delta>0$, so $L(x,y)=x^{u}y^{v}(\frac{a_{0}}{v}\ x^{2}+\frac{a_{1}}{v+1}xy+\frac{b_{2}}{u}\ y^{2})$ as in Lemma 2.1. The case $\Delta<0$ goes likewise. Fix $\eta$ such that $(\xi(\eta,T),\eta)\in\overline{\phi^{-1}(Q)\setminus Q}$. For simplicity of notation, we will suppress the $\eta$ and $T$ in $\xi(\eta,T)$. We use the variable $M=y/x$, so $y=Mx$ and differentiating gives $\dot{y}=\dot{M}x+M\dot{x}$. Recalling that $c_{i}=a_{i}+b_{i}$ and inserting the values for $\dot{x}$ and $\dot{y}$ from (3), we get

[TABLE]

Assume that we are in the level set $L(x,y)=L(\xi,\eta)=\xi^{u}\eta^{v}(\frac{a_{0}}{v}\xi^{2}+\frac{a_{1}}{v+1}\xi\eta+\frac{b_{2}}{u}\eta^{2})$, then we can solve for $x^{2}$ in the expression

[TABLE]

Here we used

[TABLE]

(where the last step follows from (4)) and a similar computation for the term with $x,y$.

Use (10) and (11) to obtain

[TABLE]

This gives

[TABLE]

where recall $\beta_{0}=\frac{u+v+2}{2v}$ and $\beta_{2}=\frac{u+v+2}{2u}$ from (11), which also gives $1-\frac{2}{u+v+2}=\frac{1}{2\beta_{0}}+\frac{1}{2\beta_{2}}$. Combined with (13), this gives

[TABLE]

with

[TABLE]

For the exit time $T\geq 0$, recall that $\xi(\eta,T)$ and $\omega(\eta,T)$ are such that the solution of (3) satisfies $(x(0),y(0))=(\xi(\eta,T),\eta)$ and $(x(T),y(T))=(\zeta_{0},\omega(\eta,T))$. This implies $M(0)=\eta/\xi(\eta,T)$ and $M(T)=\omega(\eta,T)/\zeta_{0}$. Inserting this in (16), separating variables, and integrating we get

[TABLE]

In the rest of the proof, we will frequently suppress the dependence on $\eta$ and $T$ in $\xi(\eta,T)$ and $\omega(\eta,T)$. We know that $L(\xi(\eta,T),\eta)=\xi^{u}\eta^{v}(\frac{a_{0}}{v}\xi^{2}+\frac{b_{2}}{u}\eta^{2})=\zeta_{0}^{u}\omega^{v}(\frac{a_{0}}{v}\zeta_{0}^{2}+\frac{b_{2}}{u}\omega^{2})=L(\eta,\omega(\eta,T))$, which gives

[TABLE]

From their definition, $\xi(\eta,T)$ and $\omega(\eta,T)$ are clearly decreasing in $T$, so their $T$-derivatives $\xi^{\prime}(\eta,T),\omega^{\prime}(\eta,T)\leq 0$. Since $c_{0},c_{2}>0$ (otherwise $\Delta=0$), the integrand of (18) is $O(M^{\frac{1}{\beta_{0}}-1})$ as $M\to 0$ and $O(M^{-\frac{1}{\beta_{2}}-1})$ as $M\to\infty$. Hence the integral is increasing and bounded in $T$. But this means that $G(\xi(\eta,T),\eta)T$ is increasing in $T$ and bounded as well. Let $g(\eta,T)=\xi(\eta,T)T^{\beta_{2}}$. Since

[TABLE]

and $1-\frac{1}{2\beta_{0}}-\frac{1}{2\beta_{2}}>0$, we find that $g(\eta,T)$ converges222For the symmetric statement on $\omega(\eta,T)$, define $\hat{g}(\eta,T)=\omega(\eta,T)T^{\beta_{0}}$. Then $\lim_{T\to\infty}\hat{g}(\eta,T)=\lim_{T\to\infty}g(\eta,T)^{\beta_{0}/\beta_{2}}\eta^{1+2/v}\zeta_{0}^{-b_{0}/a_{0}}(\frac{c_{2}}{c_{0}})^{1/v}$.:

[TABLE]

where we have used $-\beta_{2}(1-\frac{1}{2\beta_{0}}-\frac{1}{2\beta_{2}})=-\frac{2\beta_{2}}{u+v+2}=-\frac{1}{u}$ for the exponent of $c_{2}$, and $\frac{2}{u}+\frac{\beta_{2}}{\beta_{0}}=\frac{v+2}{u}=\frac{a_{2}}{b_{2}}$ for the exponent of $\eta$.

We continue the proof to get higher asymptotics. Differentiating (18) w.r.t. $T$ gives

[TABLE]

where (by differentiating (17))

[TABLE]

Combined with (17), (19) and (21), this gives

[TABLE]

Because $\frac{1}{2\beta_{0}}+\frac{1}{2\beta_{2}}-1=-\frac{2}{u+v+2}$, using (11) and dividing by $\eta^{\frac{1}{\beta_{0}}}$, we can simplify (2) to

[TABLE]

Taking the derivative of (19) w.r.t. $T$ and multiplying with $\Delta/(c_{0}c_{2})$ gives

[TABLE]

Hence, we can rewrite (23) as

[TABLE]

We insert $\xi^{\prime}=g^{\prime}(T)T^{-\beta_{2}}-\beta_{2}g(T)T^{-(1+\beta_{2})}$ and multiply with $T^{\beta_{2}}$, which leads to

[TABLE]

Since $\xi=O(T^{-\beta_{2}})$ and $\omega=O(T^{-\beta_{0}})$, we can write this differential equation as

[TABLE]

Keeping the leading terms only (where we use that $2\beta_{2},2\beta_{0}>1$), we get the differential equation

[TABLE]

Using the limit boundary value $\xi_{0}=\xi_{0}(\eta)=\lim_{T\to\infty}g(\eta,T)$, we find the solution

[TABLE]

as required. The analogous asymptotics for $\omega$ and the constants $\omega_{0}$ and $\omega_{1}$ can be derived by changing the time direction and the roles $(a_{0},a_{2})\leftrightarrow(b_{2},b_{0})$, and also by the relation $\xi^{u}\eta^{v+2}c_{2}\sim\zeta_{0}^{u+2}\omega^{v}c_{0}$ from (19):

[TABLE]

This concludes the proof. $\Box$

3 Proof of Theorem 1.1

To prove that the regular variation established in Proposition 2.1 is robust under perturbations of the vector field, we put the ${\mathcal{O}}(4)$ terms back into (3), but since we consider it as a perturbation of (9),, we write $\tilde{X}$ instead:

[TABLE]

so that $|\tilde{X}-X|={\mathcal{O}}(4)$. The quantities $\xi(\eta,T),\omega(\eta,T)$ will be written as $\tilde{\xi}(\eta,T),\tilde{\omega}(\eta,T)$ etc., and the goal is to show that $\tilde{\xi}(\eta,T)$ is still regularly varying. Let us now give the proof of Theorem 1.1.

Proof. As before, let $\xi=\xi(\eta,T)$ be such that for the unperturbed flow, $\phi^{T}(\xi,\eta)=(\zeta_{0},\omega(\eta,T))$. Proposition 2.1 gives the asymptotics of $\xi(\eta,T)$ as $T\to\infty$. At the same time, under the perturbed flow associated to (24), $\phi^{\tilde{T}}(\xi,\eta)=(\zeta_{0},\tilde{\omega}(\eta,\tilde{T}))$ for some $\tilde{T}$. Therefore we can write $\xi(\eta,T)=\tilde{\xi}(\eta,\tilde{T})$, and once we estimated $\tilde{T}$ as function of $T$, we can express $\tilde{\xi}(\eta,\tilde{T})$ explicitly as function of $\tilde{T}$. We follow the argument of the proof of Proposition 2.1, keeping track of the effect of the higher order terms.

The perturbed first integral: To start, we construct a first integral $\tilde{L}$ on $Q=[0,\zeta_{0}]\times[0,\eta_{0}]$ by defining

[TABLE]

for $0<\delta\leq\min\{\zeta_{0},\eta_{0}\}$ and $t\in{\mathbb{R}}$. (We continue the argument for the case $\Delta>0$; the other case goes analogously.)

By construction, $\tilde{L}$ is constant on integral curves of $\dot{z}=\tilde{X}(z)$. Because $\tilde{X}$ is $C^{3}$, the integral curves are $C^{3}$ curves, and form a $C^{3}$ foliation of $P_{0}$, see e.g. [16, Theorem 2.10]. Note that the coordinate axes consist of the stationary point $(0,0)$ and its stable and unstable manifold; we put $\tilde{L}(x,0)=\tilde{L}(0,y)=0$. Then $\tilde{L}$ is continuous on $Q$ and $C^{2+1}$ on the interior of $Q$.

Now we compare $\tilde{L}$ with $L$ on a small neighbourhood $U$ of $\phi_{hor}^{-1}(Q)\cup Q\cup\phi_{hor}^{1}(Q)$. Take $y_{0}=\eta_{0}$ and $x_{0}=x_{0}(\delta)$ such that the integral curve of $\dot{z}=X(z)$ through $z_{0}:=(x_{0},y_{0})$ intersects the diagonal at $(\delta,\delta)$. Then the integral curve of $\dot{z}=\tilde{X}(z)$ through $z_{0}$ intersects the diagonal at $(\tilde{\delta},\tilde{\delta})$ for some $\tilde{\delta}=\tilde{\delta}(\delta)$, see Figure 3.

Therefore

[TABLE]

Estimating $\tilde{\delta}/\delta$: Parametrise the integral curve of $X$ through $z_{0}$ as $(x(y),y)$ for $\min\{\delta,\tilde{\delta}\}\leq y\leq y_{0}$. (So $x\leq y$; the case $y\leq x$ can be dealt with by switching the roles of $x$ and $y$.) Then by (3):

[TABLE]

For the perturbed vector field (24) we parametrise the integral curve of through $z_{0}$ as $(\tilde{x}(y),y)$ and we have the analogue of (26):

[TABLE]

Since $x\leq y$, the $O$-terms can be written as $O(y^{3})$. Combining (26) and (27) we obtain

[TABLE]

We will neglect the term $o(|(\tilde{x},y)|)$ because they can be absorbed in the big-$O$ terms at the end of the estimate. Integration over $[\delta,y_{0}]$ gives

[TABLE]

Since $\tilde{x}(y_{0})=x(y_{0})=x_{0}$ and $x(\delta)=\delta$, this simplifies to

[TABLE]

We solve for $x$ from $x^{u}y^{v}(\frac{a_{0}}{v}x^{2}+\frac{a_{1}}{v+1}xy+\frac{b_{2}}{u}y^{2})=L(x,y)=L(\delta,\delta)=\delta^{u+v+2}(\frac{a_{0}}{v}+\frac{a_{1}}{v+1}+\frac{b_{2}}{u})$:

[TABLE]

In particular,

[TABLE]

Combine the first two factors of (29) to

[TABLE]

Note that $\lim_{y\to\delta}U(y)=1$, and $U(y)$ is differentiable. Using (26) and (29) we compute the derivative

[TABLE]

Next we integrate by parts (assuming first that $\frac{a_{2}}{b_{2}}\neq 1$):

[TABLE]

Since $\frac{1}{b_{2}+b_{0}}\leq\frac{U(y)^{3}}{b_{2}+b_{0}(\frac{x}{y})^{2}}\leq\frac{1}{b_{2}}(1+\frac{c_{0}}{c_{2}})^{\frac{3}{u}}$ and $\frac{U(y)^{3}}{b_{2}+b_{0}(\frac{x}{y})^{2}}\to\frac{1}{b_{2}+b_{0}}$ as $y\to\delta$, there are constants $\hat{C}_{1},\hat{C}_{2}\in{\mathbb{R}}$ such that the final term in the above expression is

[TABLE]

For the case $\frac{a_{2}}{b_{2}}=1$, a similar computation gives

[TABLE]

for some generically nonzero $\hat{C}_{3},\hat{C}_{4},\hat{C}_{5},\hat{C}_{6}\in{\mathbb{R}}$.

By (27), the derivative $\tilde{x}^{\prime}(\delta)=\frac{a_{0}+a_{1}+a_{2}}{b_{0}+b_{1}+b_{2}}+O(\delta)$. Since $\tilde{\delta}$ lies between $\delta$ and $\tilde{x}(\delta)$ (see Figure 3), we have

[TABLE]

Later in the proof we need the quantity

[TABLE]

Writing $|\tilde{\delta}-\delta|$ in terms of $|\tilde{x}(\delta)-\delta|$ using (30), and combining with the above estimates for $|\tilde{x}(\delta)-\delta|$, we find

[TABLE]

for (generically nonzero) constants $C_{1},C_{2},C_{3},C_{4}\in{\mathbb{R}}$ and $C_{\log}$ is only nonzero if $\frac{a_{2}}{b_{2}}=1$.

For the region $\{x\geq y\}$ (containing the point $(x_{1},y_{1}):=(\zeta_{0},\tilde{\omega}(\eta,\tilde{T}))$) we reverse the roles $a_{2},b_{2},x_{0},y_{0}\leftrightarrow b_{0},a_{0},y_{1},x_{1}$. This gives

[TABLE]

for (generically nonzero) constants $\hat{C}_{1},\hat{C}_{2},\hat{C}_{3},\hat{C}_{4}\in{\mathbb{R}}$ and $\hat{C}_{\log}$ is only nonzero if $\frac{b_{0}}{a_{0}}=1$. Combining with (31) gives

[TABLE]

Estimate of $\tilde{T}$: Now let $z_{0}=(x_{0},y_{0})=(\xi(\eta,T),\eta)=(\tilde{\xi}(\eta,\tilde{T}),\eta)$ be the point such that $\phi_{hor}^{T}(z_{0})=(\zeta_{0},\omega(\eta,T))$ under the unperturbed flow and $\tilde{\phi}_{hor}^{\tilde{T}}(z_{0})=(\zeta_{0},\tilde{\omega}(\eta,\tilde{T}))$ under the perturbed flow. We estimate $\tilde{T}$ in terms of $T$.

Combining the estimate for $\xi(\eta,T)$ from Proposition 2.1 with $L(\delta,\delta)=L(\xi(\eta,T),\eta)$, we can find the relation between $\delta$ and $T$:

[TABLE]

for $\delta_{0}=\xi_{0}^{\frac{1}{2\beta_{2}}}\eta^{1-\frac{1}{2\beta_{2}}}(\frac{c_{2}}{c_{0}+c_{2}})^{\frac{1}{u+v+2}}$.

For $M=y/x$, computations analogous to (13) show that there is $\Psi=\Psi(x,M)=O(1+M^{3})$ such that

[TABLE]

For every $(x,y)=(x,xM)$ on the $\tilde{\phi}$-trajectory of $z_{0}$ (i.e., level set of $\tilde{L}$), we have

[TABLE]

This gives the analogue of [2, formula (32)]

[TABLE]

where $G(\xi,\eta)$ is as in (17). To estimate $\tilde{T}$, we take some increasing function $\delta\leq\rho(\delta)\leq\delta^{1/2}$ such that $\delta=o(\rho(\delta))$ and divide the trajectory $\tilde{\phi}^{t}(z_{0})=(\tilde{x}(t),\tilde{y}(t))$ of $z_{0}$ into three parts separated by two points in time:

[TABLE]

and let $T_{1},T_{2}$ be the analogous quantities for the unperturbed trajectory. We compute

[TABLE]

Similarly, using $\tilde{x}(y)/x(y)=1+O(\psi(\delta))$ as in (29),

[TABLE]

This gives

[TABLE]

by a similar computation for $T-T_{2}=\int_{\rho(\delta)}^{\zeta_{0}}\frac{dx}{\dot{x}}$, etc.

Finally, for $\tilde{T}_{1}<t<\tilde{T}_{2}$, we have $\psi(x,y)=O(\delta^{\alpha_{*}},\delta\log(1/\delta))$ by (32), and $x\Psi(x,M)=O(x+yM^{2})=(1+M^{2})O(\rho(\delta))$. Therefore

[TABLE]

Choosing $\rho(\delta)=\delta\log(1/\delta)$, and using (33) gives

[TABLE]

Combining this with (36) gives $\tilde{T}=T(1+O(T^{-\beta_{*}},T^{-\frac{1}{2}}\log T))$ for $\beta_{*}=\frac{1}{2}\min\{1,\frac{a_{2}}{b_{2}},\frac{b_{0}}{a_{0}}\}$. The estimate of Proposition 2.1 now gives $\tilde{\xi}(\eta,\tilde{T})=\xi_{0}(\eta)\tilde{T}^{-\beta_{2}}(1+O(\tilde{T}^{-\beta_{*}},\tilde{T}^{-\frac{1}{2}}\log\tilde{T}))$ as claimed.

Reversing the roles $(a_{0},a_{2})\leftrightarrow(b_{2},b_{0})$ as in the end of the proof of Proposition 2.1 gives $\tilde{\omega}(\eta,\tilde{T})=\omega_{0}(\eta)\tilde{T}^{-\beta_{0}}(1+O(\tilde{T}^{-\beta_{*}},\tilde{T}^{-\frac{1}{2}}\log\tilde{T}))$. $\Box$

The formula (6) for the Dulac maps follows directly from Theorem 1.1 by inverting $T\mapsto\xi(\eta,T)$ and inserting this in the formula for $\omega(\eta,T)$. In the special case that $\beta_{0}=\beta_{2}$, formula (6) reduces to

[TABLE]

Reducing further by assuming (5) (i.e., in the volume preserving setting), we get

[TABLE]

This coefficient $\frac{b_{2}}{a_{0}}\left(\frac{\eta}{\zeta_{0}}\right)^{3}=\frac{\|X_{hor}(0,\eta)\|}{\|X_{hor}(\zeta_{0},0)\|}$ agrees with the fact that for $\omega=D(\xi)$, the flow-boxes $\cup_{t\in[0,\varepsilon]}\phi_{hor}^{t}([0,\xi]\times\{\eta\})$ and $\cup_{t\in[0,\varepsilon]}\phi_{hor}^{t}(\{\zeta_{0}\}\times[0,\omega])$ must have the same volume. If the neutral saddle $p$ is part of a heteroclinic cycle, then it is accumulated by periodic solutions, but these are not limit cycles of course.

4 Time-$1$ map versus Poincaré map

First we give an estimate of observables integrated over the flow-lines of $X_{hor}$ of (3).

Proposition 4.1

Let $r=\sqrt{x^{2}+y^{2}}$, $\rho>0$ and $W(T)$ be the integral curve for (3) connecting $(\xi(\eta_{0},T),\eta_{0}))$ to $(\zeta_{0},\omega(\eta_{0},T)$, see Figure 1. Then there is a constant $C=C(\rho)>0$ such that

[TABLE]

Proof. We build on the proof of Proposition 2.1 (or in fact Theorem 1.1), and in the integral $\Psi$ we change coordinates $M=y/x$. That is, $r^{\rho}=(x^{2}+y^{2})^{\rho/2}=x^{\rho}(1+M^{2})^{\rho/2}$. Use (15) to get

[TABLE]

with $G(T):=G(\xi(\eta_{0},T)),\eta_{0})$ as in (17). Abbreviate $\xi(\eta_{0},T)=\xi(T)$ and $\omega(\eta_{0},T)=\omega(T)$. Inserting the above in the integral of (18), we obtain

[TABLE]

For $M\to 0$, the leading term in the integrand is

[TABLE]

i.e., the exponent is $>-1$ for $\rho<2$. For $M\to\infty$, the leading term in the integrand is

[TABLE]

i.e., the exponent is $<-1$ for $\rho<2$. This means that the integral in (38) converges to some constant $C_{0}=C_{0}(\rho)$ as $T\to\infty$, and $\Psi\sim C_{0}G(T)^{\frac{\rho}{2}-1}\sim CT^{1-\frac{\rho}{2}}$ for $C=C_{0}\left(c_{2}^{1-\frac{1}{2\beta_{0}}-\frac{1}{2\beta_{2}}}\xi_{0}^{\frac{1}{\beta_{0}}}\eta_{0}^{1-\frac{1}{\beta_{2}}}\right)^{1-\frac{\rho}{2}}$. This finishes the proof for $\rho<2$.

If $\rho>2$, then the value of $\Psi$ based on the leading terms of the integrand only, is

[TABLE]

Insert the values of $\xi(T)$ and $\omega(T)$ from Proposition 2.1 as well as the leading term of $G(T)$:

[TABLE]

The powers of $T$ cancel in this expression, proving the case $\rho>2$. Finally, if $\rho=2$, then the factor $G(T)^{\frac{\rho}{2}-1}$ in (38) disappears and the leading terms in the integrand (both as $M\to 0$ and $M\to\infty$), are $c_{0}^{-1}M^{-1}$. This gives, due to Proposition 2.1,

[TABLE]

$\Box$

The $3$-dimensional time-$1$ map $\phi^{1}$ preserves no $2$-dimensional submanifold of ${\mathcal{M}}$. Yet in order to model $\phi^{t}$ as a suspension flow over a $2$-dimensional map, we need a genuine Poincaré map. For this we choose a section $\Sigma$ transversal to $\Gamma$ and containing a neighbourhood $U$ of $p$. As an example, $\Sigma$ could be ${\mathbb{T}}^{2}\times\{0\}$, and the Poincaré map to ${\mathbb{T}}^{2}\times\{0\}$ could be (a local perturbation of) Arnol’d’s cat map; in this case (and most cases) ${\mathcal{M}}$ is not homeomorphic to ${\mathbb{T}}^{3}$ because the homology is more complicated, see [1, 13].

Let $h:\Sigma\to{\mathbb{R}}^{+}$, $h(q)=\min\{t>0:\phi^{t}(q)\in\Sigma\}$ be the first return time. Assuming that $\sup_{\Sigma}|w(x,y)|<1$, the first return time $h$ is bounded and bounded away from zero, say $0<\inf_{\Sigma}h<\sup_{\Sigma}h$.

The Poincaré map $f:=\phi^{h}:\Sigma\to\Sigma$ has a neutral saddle point $p$ at the origin. Its local stable/unstable manifolds are $W^{s}_{loc}(p)=\{0\}\times(-\varepsilon,\varepsilon)$ and $W^{u}_{loc}(p)=(-\varepsilon,\varepsilon)\times\{0\}$. Because the flow $\phi^{t}$ is a perturbation of an Anosov flow, and $f$ is a Poincaré map, it has a finite Markov partition $\{P_{i}\}_{i\geq 0}$ and we can assume that $p$ is in the interior of $P_{0}$. In the sequel, let $U$ be a neighbourhood of $p$ that is small enough that (1) is valid on $U\times[0,1]$ but also that $f(U)\supset\hat{P}_{0}\cup P_{0}$.

In order to regain the hyperbolicity lacking in $f$, let

[TABLE]

be the first return time to $Y:=\Sigma\setminus P_{0}$. Then the Poincaré map $F=f^{r}=\phi^{\tau}$ of $\phi^{t}$ to $Y\times\{0\}$ is hyperbolic, where

[TABLE]

is the corresponding first return time.

Consequently, the flow $\phi^{t}:{\mathcal{M}}\times{\mathbb{R}}\to{\mathcal{M}}$ can be modeled as a suspension flow on $Y^{\tau}=\left(\bigcup_{q\in Y}\{q\}\times[0,\tau(q))\right)/(q,\tau(q))\sim(F(q),0)$. Since the flow and section $Y\times\{0\}$ are $C^{1}$ smooth, $\tau$ is $C^{1}$ on each piece $\{r=k\}$.

Lemma 4.1

In the notation of Proposition 4.1 with $\theta=w$, we have $\tau(q)=\hat{\tau}(q)+O(1)$ and $r=\hat{\tau}(q)+\Theta(\hat{\tau}(q))+O(1)$.

Proof. By the definition of $\hat{\tau}$ we have $\phi^{\hat{\tau}}_{hor}(q)\in\hat{W}^{s}$. Therefore it takes a bounded amount of time (positive or negative) for $\phi^{\hat{\tau}}(q,0)$ to hit $Y\times\{0\}$, so $|\tau(q)-\hat{\tau}(q)|=O(1)$.

If in (37) we set $\theta=w$, then $\hat{\tau}(q)+\Theta(\hat{\tau}(q))$ indicates the vertical displacement under the flow $\phi^{t}$. In particular, it gives the number of times the flow-line intersects $\Sigma$, and hence $r=\hat{\tau}(q)+\Theta(\hat{\tau}(q))+O(1)$. $\Box$

Assume that $\phi^{t}$ and $\hat{f}$ preserve Lebesgue measure.

Proposition 4.2

Recall that $\beta_{2}=\frac{a_{2}+b_{2}}{2b_{2}}\in(\frac{1}{2},\infty)$. There exists $C^{*}>0$ such that

[TABLE]

for the $F$-invariant SRB-measure $\mu_{\bar{\phi}}$.

Proof. The function $\tau$ is defined on $\Sigma\setminus P_{0}$ and $\tau\geq h_{2}=h+h\circ f$ on $Y_{\{r\geq 2\}}:=f^{-1}(P_{0})\setminus P_{0}$. The set $Y_{\{r\geq 2\}}$ is a rectangle with boundaries consisting of two stable and two unstable leaves of the Poincaré map $f$. Let $W^{u}(y)$ denote the unstable leaf of $f$ inside $Y_{\{r\geq 2\}}$ with $(0,y)$ as (left) boundary point. Let $y_{1}<y_{2}$ be such that $W^{u}(y_{1})$ and $W^{u}(y_{2})$ are the unstable boundary leaves of $Y_{\{r\geq 2\}}$.

The unstable foliation of $\hat{f}=\phi_{hor}^{1}$ does not entirely coincide with the unstable foliation of $f$. Let $\hat{W}^{u}(y)$ denote the unstable leaf of $\hat{f}$ with $(0,y)$ as (left) boundary point. Both $\hat{W}^{u}(y)$ and $W^{u}(y)$ are $C^{1}$ curves emanating from $(0,y)$; let $\gamma(y)$ denote the angle between them. Then the lengths

[TABLE]

as $t\to\infty$, where the last equality and the notation $\xi_{0}(y)$ and $\beta_{2}=(a_{2}+b_{2})/(2b_{2})$ come from Theorem 1.1

We decompose Lebesgue on $Y_{\{r\geq 2\}}$ as

[TABLE]

The conditional measures $\mu_{W^{u}(y)}$ on $W^{u}(y)$ equals $1$-dimensional Lebesgue $m_{W^{u}(y)}$ on $W^{u}(y)$ Therefore, as $t\to\infty$,

[TABLE]

for $C^{*}=\int_{y_{1}}^{y_{2}}|\cos\gamma(y)|\ \xi_{0}(y)\,d\nu^{u}(y)$. This proves the result. $\Box$

5 The proof of Corollary 1.1

Proof. The proof of Corollary 1.1 is a direct application of Theorem 2.7 in [3], where $\bar{v}=\int_{0}^{\tau}v\circ\phi^{t}\,dt$ takes the role of $\bar{\psi}$ in [3, Theorem 2.7], but the condition that $\bar{\psi}=C-\psi_{0}$ for some positive $\psi_{0}$ is only important for the results on the shape of the pressure function in [3]. For us, only the tail of $\bar{v}$ matters and since $v$ is $C^{1}$ on ${\mathcal{M}}\setminus\Gamma$, $\bar{v}$ is $C^{1}$ on each partition element $\{\phi=n\}$ of the Markov map $F$. Since Proposition 4.1 applies to $v$ we get $\bar{v}(x,y)\sim C_{p}T^{1-\frac{\rho}{2}}$ if the Dulac time of $(x,y)$ is $T$. Since our invariant measure is Lebesgue, and $\beta_{2}=2$, Theorem 1.1 can be immediately used to estimate

[TABLE]

where $\Sigma$ is the Poincaré section and $\mbox{Vol }(\Sigma\setminus P_{0})$ is the normalizing constant for Lebesgue restricted to the domain $\Sigma\setminus P_{0}$ of $F$. If $\rho\geq 2$, this asymptotic formula should be interpreted as $\mbox{Leb}(\bar{v}>t)=0$ for $t$ large, that is: $\bar{v}$ is bounded.

The exponent of this tail is $-2$ if and only if $\rho=0$, and in this case [3, Theorem 2.7(a)(ii)] gives the non-Gaussian CLT.

If $-2<\rho<0$, [3, Theorem 2.7(a)(i)] gives a Stable Law of order $4/(2-\rho)\in(1,2)$.

Finally, if $0<\rho<2$ (or $\rho\geq 2$ when $\bar{v}$ is bounded), then we obtain the CLT provided the variance $\sigma^{2}>0$, and this follows from $\bar{v}$ not being a coboundary. In other words, $\bar{v}\neq h-h\circ F$ for any $h\in{\mathcal{B}}$, the Banach space used in the proofs of [3], and this we assumed explicitly. $\Box$

Acknowledgements: We gratefully acknowledge the support of FWF grant P31950-N45.