Entropy and Poincaré recurrence from a geometrical viewpoint

Details Entropy and Poincaré recurrence from a geometrical viewpoint Entropy and Poincaré recurrence from a geometrical viewpoint

2008-09-17

arxiv.org/abs/0809.2938v2

Mathematics

Dynamical Systems

11 pages, revised version

Scientific paper

We study Poincar\'e recurrence from a purely geometrical viewpoint. We prove that the metric entropy is given by the exponential growth rate of return times to dynamical balls. This is the geometrical counterpart of Ornstein-Weiss theorem. Moreover, we show that minimal return times to dynamical balls grow linearly with respect to its length. Finally, some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures are given.

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