## Institute for Mathematical Sciences

## Preprint ims06-03

** A. A. Pinto, D. A. Rand**
*Geometric measures for hyperbolic sets on surfaces*

Abstract: 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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