Counterexamples to double recurrence for non-commuting deterministic transformations

TL;DR

This paper demonstrates that double recurrence properties fail for certain non-commuting, mixing, zero entropy transformations when using specific polynomial iterates, answering a longstanding open question.

Contribution

It provides the first comprehensive counterexamples to double recurrence for non-commuting transformations with polynomial iterates of degree at least one.

Findings

Double recurrence fails for non-commuting, mixing, zero entropy transformations.

Counterexamples are constructed for polynomial iterates of degree 1 or ≥2.

Answers a question posed by Frantzikinakis and Host.

Abstract

We show that if $p_1,p_2$ are injective, integer polynomials that vanish at the origin, such that either both are of degree $1$ or both are of degree $2$ or higher, then double recurrence fails for non-commuting, mixing, zero entropy transformations. This answers a question of Frantzikinakis and Host completely.