Entropy and Poincaré recurrence from a geometrical viewpoint

We study Poincare recurrence from a purely geometrical viewpoint. In ´ (Downarowicz and Weiss 2004 Illinois J. Math. 48 59–69) it was proven that the metric entropy is given by the exponential growth rate of return times to dynamical balls. Here we use combinatorial arguments to provide an alternative and more direct proof of this result and to prove that minimal return times to dynamical balls grow linearly with respect to its length. Some relations using weighted versions of recurrence times are also obtained for equilibrium states. Then we establish some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures.