Abstract

Abstract

Abstract
We describe a construction of a moduli space of solenoid functions for the C 1 +-conjugacy classes of hyperbolic dynamical systems f on surfaces with hyperbolic basic sets ?f. We explain that if the holonomies are sufficiently smooth then the diffeomorphism f is rigid in the sense that it is C 1 +conjugate to a hyperbolic affine model. We present a moduli space of measure solenoid functions for all Lipschitz conjugacy classes of C 1 +- hyperbolic dynamical systems f which have a invariant measure that is absolutely continuous with respect to Hausdorff measure. We extend Livšic and Sinai’s eigenvalue formula for Anosov diffeomorphisms which preserve an absolutely continuous measure to hyperbolic basic sets on surfaces which possess an invariant measure absolutely continuous with respect to Hausdorff measure. Introduction We say that (f, ?) is a C 1 +hyperbolic diffeomorphism if it has the following properties: (i) f: M ? M is a C 1 + adiffeomorphism of a compact surface M with respect to a C 1 + astructure on M, for some a > 0. (ii) ? is a hyperbolic invariant subset of M such that f|? is topologically transitive and ? has a local product structure. We allow both the case where ? = M and the case where ? is a proper subset of M. If ? = M then f is Anosov and M is a torus. Examples where ? is a proper subset of M include the Smale horseshoes and the codimension one attractors such as the Plykin and derived-Anosov attractors. © Mathematical Sciences Research Institute 2007.

Abstract

Abstract

Abstract