Solenoid functions for hyperbolic sets on surfaces

Abstract

We describe a construction of a moduli space of solenoid functions for the C1C-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 C1C conjugate to a hyperbolic affine model. We present a moduli space of measure solenoid functions for all Lipschitz conjugacy classes of C1C- hyperbolic dynamical systems f which have a invariant measure that is absolutely continuous with respect to Hausdorff measure. We extend Livˇsic and Sinai's eigenvalue formula for Anosov diffeomorphisms which preserve an absolutely continuousmeasure to hyperbolic basic sets on surfaces which possess an invariant measure absolutely continuous with respect to Hausdorff measure.