Solenoid functions for hyperbolic sets on surfaces

• Type: Chapter or Part of a Book
• Date: 2007

Authors

Alberto A. Pinto (Author) FCUP; Rand, DA (Author) Other

Book

Title: Dynamics, Ergodic Theory, and Geometry Dedicated to Anatole Katok

Pages: 145-178

ISBN: 9780511755187; 9780521875417

Electronic ISBN: 9780521875417

Indexing

Scopus - 9 Citations

Other information

Authenticus ID: P-00A-AAY

DOI: 10.1017/cbo9780511755187.007

Abstract (EN): 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 Liv¨ic 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 + ¿diffeomorphism of a compact surface M with respect to a C 1 + ¿structure on M, for some ¿ > 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.

Language: English

Type (Professor's evaluation): Scientific