Applications of the Central Limit Theorem in dynamical systems

Question

There are very many papers in the area of (possibly non-uniformly) hyperbolic dynamical systems whose aim is to prove the Central Limit Theorem. In a dynamical context, this means that one:

• has a metric space $\Omega$, a map (or flow, but let's stick to maps) $T:\Omega\to\Omega$ and a (Borel) invariant probability measure $\mu$,

• and aims at finding a suitable space $\mathcal{H}$ of functions $f:\Omega\to \mathbb{R}$ such that for all $f\in \mathcal{H}$, denoting by $X$ a random variable of distribution $\mu$, the sequence of random variables $f(X), f(T X), f(T^2 X),\dots$ satisfies the conclusion of the CLT for some variance $\sigma^2$.

I can see the beauty of this, first as it yields for a sequence of completely dependent random variables the conclusion of a Theorem most usually proved with a high degree of independence (thus enforcing the principle that deterministic chaos is tamed by time averaging); and second as it gives a strengthening of ergodicity, giving insight on the rate of equidistribution of orbits.

My question is whether there are application of this kind of CLT? By application I don't (necessarily) mean real-life or applied science application, but rather mathematical statements who do not look like they involve the CLT, but whose proof use it in an essential way.

Answer 1

These limit theorems can be useful when studying systems preserving an infinite invariant measure.

For example, Jean Pierre Conze has used it in his paper "Sur un critere de recurrence en dimension 2 pour les marches stationnaires" to get a criteria for recurrence for $\mathbb{Z}^2$ skew product extensions. See also Klaus Schmidt's "Recurrence of cocycles and stationary random walks". In general, a local central limit theorem can be used to establish multiple recurrence in systems with infinite measure.

Another nice use is in Ledrappier and Sarig's paper where they use Ratner's central limit theorem to prove rational ergodicity (a stronger form of ergodicity for an infinite measure preserving transformation) for $\mathbb{Z}^d$ covers of horocycle flows.

Another place where it is used is in homogenisation of fast-slow systems, see for example the papers of Dolgopyat and Kelly-Melbourne.