The classical Rokhlin Lemma asserts that for an aperiodic measure-preserving transformation $T$ of a probability space, one can find a "tower" of sets on which $T$ acts by translation and which covers almost all of the space. This result is a basic tool in ergodic theory and plays a crucial role in the proofs of several fundamental theorems (e.g., Dye's and Ornstein's). In the 1970s, Ornstein and Weiss extended the Rokhlin Lemma to free actions of discrete amenable groups, thus allowing many of these results to be generalized beyond the $\mathbb{Z}$-action setting.

It is natural to ask for analogous statements in topological dynamics, where an amenable group acts by homeomorphisms on a compact metrizable space. Indeed, such results proved to be extremely useful in certain topics, such as the study of C$^*$-crossed products and the theory of mean dimension. However, it turns out that the situation is significantly more subtle than in the measurable case. On the one hand, there are several different ways to formulate a topological analogue of the Ornstein–Weiss theorem, useful for different problems. On the other hand, unlike in the measurable setting, genuine difficulties arise, stemming both from the algebraic structure of the group and from the topology of the space.

In this talk, I will begin by reviewing the classical measurable results and introducing the corresponding topological formulations. I will then briefly discuss some applications (including my work on shift embeddability), and conclude by outlining the conditions under which these topological Rokhlin-type properties can be established (following, among others, work with Petrakos).