Solenoid functions for hyperbolic sets on surfaces

Summary

We describe a construction of a moduli space of solenoid functions for the C¹⁺-conjugacy classes of hyperbolic dynamical systems f on surfaces with hyperbolic basic sets Λ _f. We explain that if the holonomies are sufficiently smooth then the diffeomorphism f is rigid in the sense that it is C¹⁺ conjugate to a hyperbolic affine model. We present a moduli space of measure solenoid functions for all Lipschitz conjugacy classes of C¹⁺-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 measureto hyperbolic basic sets on surfaces which possess an invariant measure absolutely continuous with respect to Hausdorff measure.

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 [16; 33]. 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.

Let f and g be any two C¹⁺ hyperbolic diffeomorphisms with basic sets ∧_f and ∧_g, respectively. If f and g are topologically conjugate and the conjugacy has a derivative at a point with nonzero determinant, then f and g are C¹⁺ conjugate.

See definitions of topological and C¹⁺ conjugacies in Section 2.3. A weaker version of this theorem was first proved by D. Sullivan [47] and E. de Faria [8] for expanding circle maps. Theorem 1.1 follows from [13] using the results presented in [1] and in [13] which apply to Markov maps on train tracks and to nonuniformly hyperbolic diffeomorphisms.