Multiple recurrence without commutativity

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 $$ S^{n_i}x\to x,\ T^{p_1(n_i)}x\to x, \ldots,\ T^{p_d(n_i)}x\to x,\ i\to\infty.$$