Geometric measures for hyperbolic sets on surfaces

Details Geometric measures for hyperbolic sets on surfaces Geometric measures for hyperbolic sets on surfaces

2006-05-15

arxiv.org/abs/math/0605402v1

Mathematics

Dynamical Systems

59 pages, 17 figures

Scientific paper

We present a moduli space for all hyperbolic basic sets of diffeomorphisms on surfaces that have an invariant measure that is absolutely continuous with respect to Hausdorff measure. To do this we introduce two new invariants: the measure solenoid function and the cocycle-gap pair. We extend the eigenvalue formula of A. N. Livsic and Ja. G. Sinai 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. We characterise the Lipschitz conjugacy classes of such hyperbolic systems in a number of ways, for example, in terms of eigenvalues of periodic points and Gibbs measures.

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