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'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 $$\underset{k\to \infty }{\lim }\frac{{\int }_{F_k}\left(\frac{1}{k}{\sum }_{i=0}^{k-1}{T}^{i}f\right)\mathrm{d}\mu }{\mu (F_k)}=\int f\mathrm{d}\mu ?$$

Changes were made taking into account a referee's comments