Spatial-temporal differentiation theorems

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 is ergodic with respect to the homeomorphism \(T \colon X \to X\). 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 $$\lim_{k \to \infty} \frac{\int_{F_k} ( \frac{1}{k} \sum_{i = 0}^{k - 1} T^i f ) \, \, \mathrm{d} \mu}{\mu(F_k)} = \int f \, \mathrm{d} \mu \ ?$$ We establish some sufficient conditions for these sequences to converge.