Multiple recurrence without commutativity

Abstract

We study multiple recurrence without commutativity in this paper. We show that for any two homeomorphisms $T,S:X\to X$ 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 $p_1,\dots ,p_d$ vanishing at $0$, there is some subsequence $\{n_i\}$ of $\mathbb{Z}$ with $n_i\to \infty$ satisfying $$S^{n_i}x\to x,T^{p_1(n_i)}x\to x,\dots ,T^{p_d(n_i)}x\to x,i\to \infty .$$

40 pages. arXiv admin note: text overlap with arXiv:2301.07873; text overlap with arXiv:2405.11251 by other authors