Entropy rigidity for 3D conservative Anosov flows and dispersing billiards
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GAFA Geometric And Functional Analysis
ENTROPY RIGIDITY FOR 3D CONSERVATIVE ANOSOV FLOWS AND DISPERSING BILLIARDS Jacopo De Simoi, Martin Leguil, Kurt Vinhage and Yun Yang
Abstract. Given an integer k ≥ 5, and a C k Anosov flow Φ 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 Φ is C k−ε -conjugate to an algebraic flow, for ε > 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.
1 Introduction Anosov flows and Anosov diffeomorphisms are among the most well-understood dynamical systems, including the space of invariant measures, stable and unstable distributions and foliations, decay of correlations and other statistical properties. The topological classification of Anosov systems, especially diffeomorphisms, is wellunderstood in low dimensions. For flows, constructions of “exotic” Anosov flows are often built from gluing several examples of algebraic or geometric origin. Special among them are the algebraic systems, affine systems on homogeneous spaces. In the diffeomorphism case, these are automorphisms of tori and nilmanifolds. Conjecturally, up to topological conjugacy, these account for all Anosov diffeomorphisms (up to finite cover). The case of Anosov flows is quite different. Here, the algebraic models are suspensions of such diffeomorphisms and geodesic flows on negatively curved rank one symmetric spaces. There are many constructions to give new Anosov flows, all of which come from putting together geodesic flows and/or supsensions of diffeomorphisms. In particular, quite unexpected behaviors are possible, including Anosov flows on connected manifolds which are not transitive [FW80], contact flows on hyperbolic manifolds [FH13] and many other constructions combining these ideas (see, e.g., [BBY17]). When classifying Anosov systems up to C ∞ diffeomorphisms, the question becomes very different. Here, the algebraic models are believed to distinguish themselves in many ways, including regularity of dynamical distributions and thermoJacopo De Simoi is supported by the NSERC Discovery grant, Reference No. 502617-2017. Martin Leguil was supported by the ERC Project 692925 NUHGD of Sylvain Crovisier. Yun Yang is supported by a grant from the National Science Foundation (DMS-2000167).
J. DE SIMOI ET AL.
GAFA
dynamical formalism. In the case of the latter, the first such result was obtained by Katok for geodesic flows of negatively curved surfaces, where it was shown that coincidence of metric entropy with respect to Liouville measure and topological entropy implies that the surface has constant negative curvature [Kat82, Kat88]. These works lead to the following conjecture: Conjecture 1.1 (Katok Entropy Conjecture). Let (M, g) be a connected Riemannian manifold of negative curvature, and Φ be the corresponding geodesic flow. Then htop (Φ) = hμ (Φ) if and on
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