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In this paper, we extend the notion of temporo-spatial differentiationproblems to the setting of actions of more general topological groups. Theproblem can be expressed as follows: Given an action $T$ of an amenablediscrete group $G$ on a probability space $(X, \mu)$ by automorphisms, let$(F_k)_{k = 1}^\infty$ be a F{\o}lner sequence for $G$, and let $(C_k)_{k =1}^\infty$ be a sequence of measurable subsets of $X$ with positive probability$\mu(C_k)$. What is the limiting behavior of the sequence  $$\left( \frac{1}{\mu(C_k)} \int_{C_k} \frac{1}{|F_k|} \sum_{g \in F_k} f(T_gx) \mathrm{d} \mu(x) \right)_{k = 1}^\infty$$  for $f \in L^\infty(X, \mu)$? We provide some positive convergence resultsfor temporo-spatial differentiations with respect to ergodic averages overF{\o}lner sequences, as well as with respect to ergodic averages oversubsequences of the integers (e.g. polynomials), multiple ergodic averages, andweighted ergodic averages.

Temporo-spatial differentiations for actions of locally compact groups