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.