Spatial-Temporal Differentiation Theorems

TL;DR

This paper investigates the conditions under which spatial-temporal averages of functions over shrinking sets in ergodic dynamical systems converge to the expected value, extending differentiation theorems in this context.

Contribution

It establishes new differentiation theorems for ergodic systems, linking the convergence of averages over shrinking sets to the measure-theoretic integral of functions.

Findings

Convergence of averages over shrinking sets under ergodicity.

Conditions for differentiation theorems in spatial-temporal settings.

Extension of classical differentiation results to dynamical systems.

Abstract

Let $(X, \mathcal{B}, \mu, T)$ be a dynamical system where $X$ is a compact metric space with Borel $\sigma$-algebra $\mathcal{B}$, and $\mu$ is a probability measure that's ergodic with respect to the homeomorphism $T : X \to X$. We study the following differentiation problem: Given $f \in C(X)$ and $F_k \in \mathcal{B}$, where $\mu(F_k) > 0$ and $\mu(F_k) \to 0$, when can we say that $$\lim_{k \to \infty} \frac{\int_{F_k} \left( \frac{1}{k} \sum_{i = 0}^{k - 1} T^i f \right) \mathrm{d} \mu}{\mu(F_k)} = \int f \mathrm{d} \mu ? $$