# Central limit theorem versus entropy in dynamical systems context

A dynamical system $$(S^1,T, \mu)$$, $$T_* \mu=\mu$$, $$T$$ ergodic, $$S^1$$ is circle. Assume it has central limit theorem.

Want to know the relation between its measure-theoretic entropy $$h_{\mu}(T)$$ and the central limit theorem? Is there a good reference for it? Thanks!

For the measure-theoretic entropy $$h_{\mu}(T)$$, see https://en.wikipedia.org/wiki/Measure-preserving_dynamical_system#Measure-theoretic_entropy

For the central limit theorem, means: for any smooth observable $$\phi$$ on $$S^1$$ with $$\int \phi d\mu=0$$, $$\frac{\sum_{i \le n}\phi \circ T^i}{\sqrt{n}} \stackrel{d}{\longrightarrow} N(0,\sigma^2_{\phi})$$ holds where $$\sigma^2_{\phi} \ge 0$$ is a constant.

• As I see it the normal distribution arises when minimizing the differential entropy subject to a given expectation and variance, when we don't know the expectation and variance what we minimize is the relative/cross entropy (KL divergence) and we get a normal distribution with 0 expectation estimator if unbiased and variance estimator given by the Fisher information... Commented May 5, 2020 at 9:51
• ...then in CLT S converges to $N(\mu, \sigma)$ while the mle converges to $N(0,I(\mu,\sigma)^{-1})$ so I think some kind of similar play should be play between the measure entropy and the normal here Commented May 5, 2020 at 9:58
• If the CLT as you stated it holds for $\phi$, it cannot hold for $2\phi$, right? Commented May 6, 2020 at 10:04
• @Kostya I edited my question. Commented May 6, 2020 at 14:05