Lyapunov eponents and strong exponential tails for some contact Anosov flows

TL;DR

This paper establishes exponential decay rates for the measure of points in contact Anosov flows that frequently leave a large Pesin regular set, linking Lyapunov exponents with tail estimates under certain regularity conditions.

Contribution

It proves exponential tail bounds for the measure of points with frequent excursions outside a Pesin regular set in contact Anosov flows, under a regularity assumption.

Findings

Exponential decay of measure for points frequently outside the Pesin set.

Quantitative bounds involving constants C and c.

Applicability to Gibbs measures on contact Anosov flows.

Abstract

For the time-one map $f$ of a contact Anosov flow on a compact Riemann manifold $M$, satisfying a certain regularity condition, we show that given a Gibbs measure on $M$, a sufficiently large Pesin regular set $P_0$ and an arbitrary $\delta \in (0,1)$, there exist positive constants $C$ and $c$ such that for any integer $n \geq 1$, the measure of the set of those $x\in M$ with $f^k(x) \notin P_0$ for at least $\delta n$ values of $k = 0,1, \ldots,n-1$ does not exceed $C e^{-cn}$.