Spectra of Ruelle transfer operators for Axiom A flows

Abstract

For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to those of Dolgopyat (1998 Ann. Math. 147 357–90) for geodesic flows on compact surfaces (for general potentials) and transitive Anosov flows on compact manifolds with C1 jointly non-integrable horocycle foliations (for the Sinai–Bowen–Ruelle potential). Here we deal with general potentials and on spaces of arbitrary dimension, although under some geometric and regularity conditions. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, closed orbit counting functions, and in other areas.