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Some results on probability theory can be generalized to more abstract ones in ergodic theory, for example:

  • the strong law of large numbers can be seen as a particular case of Birkhoff's ergodic theorem;
  • there are some systems $T:X\to X$ that admit a convenient invariant measure $\mu$ that satisfies the Central Limit Theorem: for every function $f:X \to \mathbb{R}$ of a specific type, the limit $$ \lim_{n\to +\infty} \mu\left\{x\in X \mid \frac{1}{\sqrt{n}}\sum_{k=0}^{n-1}f\circ T^{k}(x)\leq c\right\}=\frac{1}{\sigma \sqrt{2\pi}}\int_{-\infty}^{c}e^{\frac{-x^{2}}{2\sigma^{2}}}dx,$$ holds for every $c\in \mathbb{R}$ (here $\sigma$ is a convenient constant). See this post for more details.

On the other hand, is not unusual to obtain dynamical properties of the systems when one already knows that it admits an ergodic measure, for example (if $X$ is a regular enough metric space, $T$ is continuous, and the invariant measure $\mu$ is ergodic, then $T$ is transitive when restricted to the support of the measure).

My question is: can we respect something from the dynamics if we already know that the system satisfies some form of the Central Limit Theorem?

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