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.