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 $xin M$ with $f^k(x) otin P_0$ for at least $delta n$ values of $k = 0,1, ldots,n-1$ does not exceed $C e^{-cn}$.
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