Hitting time and dimension in Axiom A systems and generic interval excanges

Abstract

In this note we prove that for equilibrium states of axiom A systems the time ${\tau }_B(x)$ needed for a typical point $x$ to enter for the first time in a typical ball $B$ with radius $r$ scales as ${\tau }_B(x)\sim r^d$ where $d$ is the local dimension of the invariant measure at the center of the ball. A similar relation is proved for a full measure set of interval excanges. Some applications to Birkoff averages of unbounded (and not $L^1$) functions are shown.