Topological mild mixing of all orders along polynomials

Yang Cao; Song Shao

doi:10.3934/dcds.2021150

Article Contents

2022, Volume 42, Issue 3: 1163-1184. Doi: 10.3934/dcds.2021150

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Topological mild mixing of all orders along polynomials

Yang Cao and
Song Shao^,

CAS Wu Wen-Tsun Key Laboratory of Mathematics, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China

^*Corresponding author: Song Shao
^*Corresponding author: Song Shao

Received: March 2021

Revised: August 2021

Early access: October 2021

Published: March 2022

This research is supported by NNSF of China (11971455, 11571335)

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Abstract

A minimal system $ (X,T) $ is topologically mildly mixing if for all non-empty open subsets $ U,V $, $ \{n\in {\mathbb Z}: U\cap T^{-n}V\neq \emptyset\} $ is an IP$ ^* $-set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that $ (X,T) $ is a topologically mildly mixing minimal system, $ d\in {\mathbb N} $, $ p_1(n),\ldots, p_d(n) $ are integral polynomials with no $ p_i $ and no $ p_i-p_j $ constant, $ 1\le i\neq j\le d $. Then for all non-empty open subsets $ U , V_1, \ldots, V_d $, $ \{n\in {\mathbb Z}: U\cap T^{-p_1(n) }V_1\cap T^{-p_2(n)}V_2\cap \ldots \cap T^{-p_d(n) }V_d \neq \emptyset \} $ is an IP$ ^* $-set. We also give the corresponding theorem for systems under abelian group actions.

Keywords:

IP system,
mild mixing,
polynomial.

Mathematics Subject Classification: Primary: 37B20; Secondary: 37B05, 37A25.

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References

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