Question

Bonjour, I need some help regarding recurrence theorems in shift spaces. I am aware of Poincaré's Recurrence Theorem, but I'm sure I've heard of another one telling about the time required to get back to a neighborhood. I appreciate any advice regarding recurrence theorems in symbolic dynamics and ergodic theory, thank you!

Answer 1

If $\mu$ is a probability measure on $X$ and $T\colon X\to X$ is an ergodic measure-preserving transformation, then for every set $A$ with positive $\mu$-measure, the Poincare Recurrence Theorem tells you that $\mu$-a.e. point $x\in A$ has finite first return time $\tau_A(x)$. There are various results giving more detailed information about the first return time, but the most general is probably Kac's formula, which states that $\int_A \tau_A(x) \,d\mu(x) = 1/\mu(A)$, and more generally that if $\phi\in L^1(X)$, then $\int_A \sum_{k=1}^{\tau_A(x)} \phi(T^k x) \,d\mu(x) = \int_X \phi(x)\,d\mu(x)$. (The result on average return time follows by taking $\phi$ to be the characteristic function of $A$).