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Topologically mildly mixing of higher orders along generalized polynomials

Yang Cao, Jianjie Zhao

Abstract

This paper is devoted to studying the multiple recurrent property of topologically mildly mixing systems along generalized polynomials. We show that if a minimal system is topologically mildly mixing, then it is mild mixing of higher orders along generalized polynomials. Precisely, suppose that $(X, T)$ is a topologically mildly mixing minimal system, $d\in \mathbb{N}$, $p_1, \dots, p_d$ are integer-valued generalized polynomials with $(p_1, \dots, p_d)$ non-degenerate. Then for all non-empty open subsets $U , V_1, \dots, V_d $ of $X$, $$\{n\in \Z: U\cap T^{-p_1(n) }V_1 \cap \dots \cap T^{-p_d(n) }V_d \neq \emptyset \}$$ is an IP$^*$-set.

Topologically mildly mixing of higher orders along generalized polynomials

Abstract

This paper is devoted to studying the multiple recurrent property of topologically mildly mixing systems along generalized polynomials. We show that if a minimal system is topologically mildly mixing, then it is mild mixing of higher orders along generalized polynomials. Precisely, suppose that is a topologically mildly mixing minimal system, , are integer-valued generalized polynomials with non-degenerate. Then for all non-empty open subsets of , is an IP-set.
Paper Structure (9 sections, 26 theorems, 149 equations)

This paper contains 9 sections, 26 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subsets of integers
  4. Partition regular sets
  5. Topological dynamics
  6. Generalized polynomials
  7. Proof of Theorem \ref{['thm general']} for degree $1$ integer-valued generalized polynomials
  8. The PET-induction and the generalized polynomial extension of van der Waerden's theorem
  9. Proof of Theorem \ref{['thm general']}

Key Result

Theorem 1.1

Let $(X, T)$ be a topologically mildly mixing minimal system, $p_1, \dots, p_d$ be integer-valued generalized polynomials with the property that $(p_1, \dots,p_d)$ is non-degenerate (see Definition def-nondegenerate). Then for all non-empty open subsets $U , V_1, \dots, V_d$ of $X$, is an IP$^*$-set.

Theorems & Definitions (56)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Claim 1
  • proof
  • ...and 46 more