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Let $T \colon X \to X$ be a continuous map on, say, a compact metric space $X$. Let $\mu$ be an invariant borel measure. Under suitable conditions, a result of Brin and Katok states that $\mu$-almost surely,

$$\lim_{\epsilon \to 0} \limsup_{n \to \infty} -\frac{1}{n}\log \mu B_x(n, \epsilon)$$

converges to an integrable function $h^*(x)$ for which $\int h^* d \mu = h_{\mu}(T)$, the metric entropy of $T$, where $$ B_x(n, \epsilon) = \{y \in X \mid d(T^i x, T^i y) \leq \epsilon, 0 \leq i \leq n\} $$ is the $(n, \epsilon)$ Bowen Ball at $x$.

I think it was Thieullen (if not, please correct me!) who introduced the notion of $\alpha$-entropy as a modification of this formula:

$$h^{\alpha}_{\mu}(x) = \lim_{\epsilon \to 0} \limsup_{n \to \infty} -\frac{1}{n} \log \mu B_x^{\alpha}(n, \epsilon) $$

where

$$B^{\alpha}_x(n, \epsilon) = \{y \in X \mid d(T^i x, T^i y) \leq e^{- i \alpha} \epsilon\}$$

I'm mostly interested in when $\mu$ is $T$-ergodic, so that $h^{\alpha}_{\mu}(x)$ is $\mu$-a.s. constant.

I've only seen this concept applied to smooth dynamical systems, but I'm curious if there are more general results.

Questions: Aside from Thieullen's papers on a generalization of the Pesin formula for the $\alpha$-entropy and the paper "Fibres dynamiques asymptotiquement compacts exposants de Lyapunov. Entropie. Dimension", are there other papers that use the $\alpha$-entropy? Is it true in the ergodic case that $h_{\mu}^{\alpha} \to h_{\mu}(T)$ as $\alpha \to 0$, and if not, what goes wrong?

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  • $\begingroup$ I'll note that Barbara Schapira was the one who suggested the edits, but from my end the LaTeX wasn't rendering correctly. In any event, she deserves the credit for the edit. $\endgroup$ – Todd Trimble Nov 1 '13 at 0:17

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