  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 $C e^{-cn}$.
