Entropy rigidity for 3D conservative Anosov flows and dispersing billiards

Abstract

Given an integer $k \ge 5$, and a $C^k$ Anosov flow $\Phi$ on some compact connected 3-manifold preserving a smooth volume, we show that the measure of maximal entropy is the volume measure if and only if $\Phi$ is $C^{k-\varepsilon }$-conjugate to an algebraic flow, for $\varepsilon >0$ arbitrarily small. Moreover, in the case of dispersing billiards, we show that if the measure of maximal entropy is the volume measure, then the Birkhoff Normal Form of regular periodic orbits with a homoclinic intersection is linear.

Bowen–Margulis Measure

In this appendix, we assume that $\Phi$ is a topologically mixing smooth Anosov flow on some compact 3-manifold M. Let us recall that by a result of Plante [Pla72], $\Phi$ is topologically mixing if and only if $E_\Phi ^s$ and $E_\Phi ^u$ are not jointly integrable. As the measure of maximal entropy is unique, it is given by the construction introduced by Margulis, which we now recall. It is first done by constructing a family of measures $\nu ^{cu}$ and $\nu ^s$ defined on leaves of the unstable foliation $\mathcal {W}_\Phi ^{u}$ and of the stable foliation $\mathcal {W}_\Phi ^s$, respectively, such that:

$$\begin{aligned} (\Phi ^t)_*\nu ^{u} = e^{ht}\nu ^{u},\qquad (\Phi ^t)_*\nu ^s = e^{-ht}\nu ^s, \end{aligned}$$

(A.1)

where $h:=h_{\mathrm {top}(\Phi )}>0$ is the topological entropy of $\Phi$. Moreover, $\nu ^{u}$ is invariant by holonomies along the leaves of $\mathcal {W}^s_\Phi$, while $\nu ^s$ is invariant by holonomies along the leaves of $\mathcal {W}^{u}_\Phi$. Notice that (A.1) allows $\nu ^u$ and $\nu ^s$ to be easily extended to measures $\nu ^{cu}$ and $\nu ^{cs}$ on leaves of the weak unstable foliation $\mathcal {W}_\Phi ^{cu}$ and of the weak stable foliation $\mathcal {W}_\Phi ^{cs}$, respectively.

The Bowen-Margulis measure $\mu$ of $\Phi$ on M is then constructed locally using the local product structure of the manifold. That is, at $x \in M$, choose open neighbourhoods of $U(x) \subset \mathcal {W}_\Phi ^{cu}(x)$ and $V(x) \subset \mathcal {W}_\Phi ^s(x)$, so that there is a well-defined map $\varphi {:}\,O(x) \rightarrow M$ which gives Hölder coordinates on the local product cube $O(x):=U(x) \times V(x)\subset M$. Fix an arbitrary $y \in U(x)$. For any open set $\Omega \subset O(x)$, we let

$$\begin{aligned} \mu (\Omega ):=\int _{z \in V(x)}\nu ^{cu}\big (U(x)\times \{z\}\cap \Omega \big ) d \nu _y(z), \end{aligned}$$

where for any $\Omega '\subset V(x)$, we set $\nu _y(\Omega '):=\nu ^{s} (\{y\}\times \Omega ')$. By the invariance of $\nu ^{s}$ under weak unstable holonomies, the previous definition is independent of the choice of $y \in U(x)$ and defines locally the Bowen-Margulis measure $\mu =\nu ^{cu} \times \nu ^s$.

Proposition A.1

If the measure of maximal entropy $\mu$ of $\Phi$ is absolutely continuous with respect to Lebesgue measure, then for any $x \in M$, $\nu ^{cu}$ is absolutely continuous with respect to Lebesgue measure on the weak unstable leaf $\mathcal {W}_\Phi ^{cu}(x)$. Furthermore, the density $e^{\psi }$ is Hölder continuous and smooth within the leaf $\mathcal {W}_\Phi ^{cu}(x)$, and

$$\begin{aligned} \psi (\Phi ^t(y)) - \psi (y) + ht = \log J^u_y(t), \quad \forall \, y \in \mathcal {W}_\Phi ^{cu}(x), \end{aligned}$$

(A.2)

where $J^u_y(t)$ is the Jacobian of the map $D \Phi ^t|_{E_\Phi ^u(y)}{:}\, E_\Phi ^u(y) \rightarrow E_\Phi ^u(\Phi ^t(y))$.

Proof

Fix $x \in M$ and choose a neighbourhood $O(x)=U(x)\times V(x)$ with local product structure and let $\varphi$ be coordinates on O(x) as described above. Fix a point $p=\varphi (y,z)\in O(x)$. On the one hand, by the construction recalled previously, we have

$$\begin{aligned} d\mu (p)=d \nu ^{cu} (y)\otimes d\nu ^s(z). \end{aligned}$$

(A.3)

On the other hand, the measure $\mu$ has local product structure, hence there exists a positive Borel function $\rho {:}\,U(x)\times V(x)$ such that

$$\begin{aligned} d\mu (p)= \rho (y,z) d \mu ^{cu}_x(y)\otimes d\mu ^s_x(z), \end{aligned}$$

(A.4)

where $\{\mu ^{cu}_q\}_{q \in O(x)}$, resp. $\{\mu ^{s}_q\}_{q \in O(x)}$ is a system of conditional measures of $\mu$ for the foliation $\mathcal {W}_\Phi ^{cu}$, resp. $\mathcal {W}_\Phi ^{s}$. As the foliations $\mathcal {W}_\Phi ^{cu}$ and $\mathcal {W}_\Phi ^{s}$ are absolutely continuous, for almost every $q \in O(x)$, the conditional measure $\mu ^{cu}_q$, resp. $\mu ^{s}_q$ is absolutely continuous with respect to the Lebesgue measure on the leaf $\mathcal {W}_\Phi ^{cu}(q)$, resp. $\mathcal {W}_\Phi ^{s}(q)$. If we fix $z \in V(x)$, we deduce from (A.3) to (A.4) and the previous discussion that

$$\begin{aligned} d\nu ^{cu} (y)= e^{\psi (y)} d y, \end{aligned}$$

(A.5)

where dy denotes the Lebesgue measure on $U(x)\subset \mathcal {W}_\Phi ^{s}(x)$. Applying $(\Phi ^t)_*$, it follows from (A.1) and (A.5) that

$$\begin{aligned} e^{ht} d \nu ^{cu}(\Phi ^t(y))= e^{ht+\psi (\Phi ^t(y))} dy=e^{\psi (y)} (\Phi ^t)_*dy=e^{\psi (y)} J^u_y(t) dy, \end{aligned}$$

(A.6)

for some measurable function $\psi {:}\,M \rightarrow {\mathbb {R}}$, where $J^u_y(t)$ is the unstable Jacobian of $D \Phi ^t$ at y. Thus, for almost every y, it holds

$$\begin{aligned} \psi (\Phi ^t(y)) - \psi (y) + ht =\log J^u_y(t). \end{aligned}$$

In other words, $\psi$ is a measurable transfer function making $\log J^u$ cohomologous to a constant. By Livsic’s theorem, $\psi$ coincides almost everywhere with a Hölder solution which is smooth along the unstable leaves and Hölder transversally. With the upgraded regularity, we define an a priori new family of conditionals along the unstable leaves at every point. By uniqueness of the family of measures satisfying (A.1) (up to multiplicative constant), these must coincide with $\nu ^{cu}$ up to fixed scalar. Therefore, $\nu ^{cu}$ is absolutely continuous with smooth density. $\square$

Remark A.2

By Proposition A.1, for any periodic point y of period $\mathcal {L}(y)>0$, taking $t=\mathcal {L}(y)$ in (A.2), we obtain another proof of Proposition 3.1 when $\Phi$ is topologically mixing. Conversely, if (3.1) holds for any periodic orbit, then by Livsic’s theorem, the $C^1$ cocycle

$$\begin{aligned} C:(y,t)\mapsto \log J^u_y(t)-ht \end{aligned}$$

over $\Phi$ is a coboundary, and (A.2) follows.