On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces

TL;DR

This paper investigates the statistical behavior of unstable curves under pseudo-Anosov diffeomorphisms on compact surfaces, showing their asymptotics are governed by cohomological action and establishing spectral properties of associated dynamical systems.

Contribution

It provides a new understanding of ergodic integrals for pseudo-Anosov maps, linking asymptotics to cohomology and proving the absence of Ruelle resonances in certain intervals.

Findings

Asymptotics of ergodic integrals are determined by cohomological action.

Toral Anosov diffeomorphisms have no Ruelle resonances in (1, e^{h_{top}}).

Results extend understanding of statistical properties of pseudo-Anosov systems.

Abstract

Weprovethattheasymptoticsofergodicintegralsalonganinvariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, are determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of Giulietti and Liverani [GL] on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval $(1,e^{h_{top}} )$.