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Entrance Time and Rényi Entropy

For ergodic systems with generating partitions, the well known result of Ornstein and Weiss shows that the exponential growth rate of the recurrence time is almost surely equal to the metric entropy. Here we look at the exponential growth rate of entrance times, and show that it equals the entropy, where the convergence is in probability in the product measure. This is however under the assumptions that the limiting entrance times distribution exists almost surely. This condition looks natural in the light of an example by Shields in which the limsup in the exponential growth rate is infinite almost everywhere but where the limiting entrance times do not exist. We then also consider $ϕ$-mixing systems and prove a result connecting the Rényi entropy to sums over the entrance times orbit segments.