Activities This Week

We introduce an ergodic optimization problem inspired by information theory, which can be presented informally as follows: given topological dynamical systems $(X, T), (Y, S)$, and a continuous function $f \in C(X \times Y)$, what can be said about the extrema $\sup_{y \in Y} \inf_{x \in X} \lim_{k \to \infty} \frac{1}{k} \sum_{j = 0}^{k - 1} f \left( T^j x , S^j y \right)?$

We will present a refinement, developed in collaboration with Bjorklund, of Furstenberg’s ergodic-theoretic approach tailored to addressing the problem of identifying ‘twisted infinite patterns’ in positive-density subsets of the integer lattice. These patterns correspond to an infinite structure within the ‘sum-products’ formed by such sets. The talk is based on joint work with Bjorklund, Bulinski, and Skinner.