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Rényi entropy and recurrence

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  • This paper studies the relationship between the return time $\tau_n$ and the Rényi Entropy Function of order $s$, $R(s)$. For a dynamical system with an invariant $\alpha$-mixing measure $\mu$ and a measurable partition, we consider the sum $W$ of measures of cylinders along orbit segments of length $\tau_n$ and relate that growth/decay rate to the R$\acute{\textrm{e}}$nyi Entropy. The key strategy is to introduce the hitting number $\nu_x(A) = | \{1 \leq i \leq \tau_n(x) : T^i(x) \in A\}|$, the number of times that $x$ hits the set $A$ when $x$ travels along its orbit of length $\tau_n(x)$, and write $W=\sum \nu_x(A) \mu(A)^s$, where the sum is taken over the $n$-cylinders. Then we show that $\nu_x(A) \approx \exp(n h_{\mu}) \mu(A)$ for most $n$-cylinders $A$. Hence $W \approx \exp(nh_{\mu}) \sum \mu(A)^{1+s}$, which relates $\tau_n(x)$ to $R(s)$, as the sum $\sum \mu(A)^{1+s} \approx \exp(-nsR(s))$.
    Mathematics Subject Classification: Primary: 37A50, 28D05; Secondary: 60E10.


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