Regular decay of ball diameters and spectra of Ruelle operators for contact Anosov flows

TL;DR

This paper investigates the decay rates of stable manifold diameters in contact Anosov flows and derives spectral estimates for Ruelle transfer operators, with implications for dynamical zeta functions and orbit counting.

Contribution

It establishes uniform decay rates of stable diameters and derives new spectral estimates for Ruelle operators in contact Anosov flows, especially geodesic flows on negatively curved manifolds.

Findings

Decay rates are similar for all small stable ball diameters.

Spectral estimates for Ruelle transfer operators are obtained.

Results apply to geodesic flows on negatively curved manifolds.

Abstract

For Anosov flows on compact Riemann manifolds we study the rate of decay along the flow of diameters of balls $B^s(x,\ep)$ on local stable manifolds at Lyapunov regular points $x$. We prove that this decay rate is similar for all sufficiently small values of $\epsilon > 0$. From this and the main result in \cite{kn:St1}, we derive strong spectral estimates for Ruelle transfer operators for contact Anosov flows with Lipschitz local stable holonomy maps. These apply in particular to geodesic flows on compact locally symmetric manifolds of strictly negative curvature. As is now well known, such spectral estimates have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, asymptotics of closed orbit counting functions, etc.