Invariants for smooth conjugacy of hyperbolic dynamical systems. IV

Abstract

We show that if two C°° transitive Anosov flows in a threedimensional manifold are topologically conjugate and the Lyapunov exponents on corresponding periodic orbits agree, then the conjugating homeomorphism is C 00 .

Smooth classification of geometrically finite one-dimensional maps

“…Sullivan [11] also proved that if the eigenvalues of f and g at all corresponding periodic points are the same, then h is smooth. Similar work has been done by Herman [3] for circle diffeomorphisms and by de la Llave and R. Moriyón [8] for Anosov diffeomorphisms of a torus. All these are results for maps without critical points.…”

Differentiability, rigidity and Godbillon-Vey classes for Anosov flows

“…The continuous invariant volume form for a transitive Anosov flow is unique up to a scalar multiple, so Θ must be volume-preserving, and hence Θ is C 3 by an application of the regularity theory in [50], and Theorem 2.6 for k = 6, and n = 2 the dimension of the transversal to the flow. Let Ψ be adapted coordinates on M, andΨ be adapted coordinates onM .…”

Book Review: Fine structure of hyperbolic diffeomorphisms

“…Finally a rigidity phenomenon was discovered by Shub and Sullivan [25] in the dynamics of expanding maps of the circle: if two such maps are conjugate by an absolutely continuous homeomorphism, then the conjugacy is smooth. A related result was obtained by de la Llave and Moryon in [10]: if the conjugacy between two Anosov diffeomorphisms of the two-dimensional torus maps periodic points into periodic points with the same eigenvalues, then the conjugacy is smooth. A better description of the smooth conjugacy classes of Anosov diffeomorphisms on the two-dimensional torus was obtained in [3].…”