Topological multiple recurrence of weakly mixing minimal systems for generalized polynomials

TL;DR

This paper proves that in weakly mixing minimal systems, the orbits generated by non-degenerate generalized polynomials are topologically recurrent and dense in the product space for a residual set of points.

Contribution

It establishes a topological multiple recurrence result for weakly mixing minimal systems using generalized polynomials, extending previous recurrence theories.

Findings

Existence of residual set with dense orbits for generalized polynomial iterates

Topological multiple recurrence in weakly mixing minimal systems

Extension of recurrence results to generalized polynomial sequences

Abstract

Let $(X, T)$ be a weakly mixing minimal system, $p_1, \cdots, p_d$ be integer-valued generalized polynomials and $(p_1,p_2,\cdots,p_d)$ be non-degenerate. Then there exists a residual subset $X_0$ of $X$ such that for all $x\in X_0$ $$\{ (T^{p_1(n)}x, \cdots, T^{p_d(n)}x): n\in \mathbb{Z}\}$$ is dense in $X^d$.