Dimension via Waiting time and Recurrence

Abstract

Quantitative recurrence indicators are defined by measuring the first entrance time of the orbit of a point $x$ in a decreasing sequence of neighborhoods of another point $y$. It is proved that these recurrence indicators are a.e. greater or equal to the local dimension at $y$, then these recurrence indicators can be used to have a numerical upper bound on the local dimension of an invariant measure.