Dimension via Waiting time and Recurrence

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 $y$. Moreover, the estimation is sharph if some mild assumptions on the statistic of return times are satisfied. These recurrence indicators can hence be used to have an efficient numerical estimation of the local dimension of an invariant measure.

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Published 1 January 2005