Extended Livšic-Sinai eigenvalue formula

We present an extension of the eigenvalue formula of A. N. Livšic and Ja. G. Sinai for Anosov diffeomorphisms that preserve an absolutely continuous measure to hyperbolic basic sets on surfaces which possess an invariant measure absolutely continuous with respect to Hausdorff measure. We also give a characterization of the Lipschitz conjugacy classes of such hyperbolic systems in a number of ways, for example following De la Llave, Marco and Moriyon, in terms of eigenvalues of periodic points and Gibbs measures.