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

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