Multiple recurrence without commutativity

TL;DR

This paper investigates multiple recurrence phenomena in topological dynamics without assuming commutativity, demonstrating residual recurrence properties for minimal homeomorphisms and nonlinear polynomial iterates.

Contribution

It establishes a residual multiple recurrence result for pairs of minimal homeomorphisms without requiring commutativity, extending classical recurrence theorems.

Findings

Residual set of points with recurrence properties

Recurrence holds along subsequences with polynomial iterates

Results apply to non-commuting minimal systems

Abstract

We study multiple recurrence without commutativity in this paper. We show that for any two homeomorphisms $T,S: X\rightarrow X$ with $(X,T)$ and $(X,S)$ being minimal, there is a residual subset $X_0$ of $X$ such that for any $x\in X_0$ and any nonlinear integral polynomials $p_1,\ldots, p_d$ vanishing at $0$, there is some subsequence $\{n_i\}$ of $\mathbb Z$ with $n_i\to \infty$ satisfying $$ S^{n_i}x\to x,\ T^{p_1(n_i)}x\to x, \ldots,\ T^{p_d(n_i)}x\to x,\ i\to\infty.$$