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.