Finitely Additive Measures on the Unstable Leaves of Anosov Diffeomorphisms

Abstract

We obtain a qualitative characterization of the convergence rate of the averages (with respect to the Margulis measure) of C2 functions over the iterations of domains in unstable manifolds of a topologically mixing C3 Anosov diffeomorphism with oriented invariant foliations. For this purpose, we extend the constructions of Margulis and Bufetov and introduce holonomy invariant families of finitely additive measures on unstable leaves and a Banach space in which holonomy invariant measures correspond to the (generalized) eigenfunctions of the transfer operator with biggest eigenvalues.