Non-recurrence and divergent $\bf ({p(n)},{q(n)})$-averages for deterministic automorphisms

TL;DR

This paper investigates the convergence of certain ergodic averages for deterministic automorphisms, showing divergence under specific conditions and providing examples of non-recurrence in systems with complex spectral properties.

Contribution

It demonstrates divergence of $(p(n),q(n))$-averages for automorphisms with particular spectral features and constructs examples of non-recurrence in mixing systems with zero entropy.

Findings

Existence of automorphisms with simple singular spectrum where averages diverge.

Construction of non-recurrence examples in systems with mixed spectral types.

Divergence of ergodic averages for sequences with unbounded differences.

Abstract

We answer the question of Frantzikinakis and Host about the convergence of ergodic $(n^2,n^3)$-averages and consider a more general case. Let sequences ${ p(n)},{ q(n)}$ satisfy the property $ p(n+1)- p(n), \ q(n+1)- q(n)\ \to\ +\infty.$ Then there exist automorphisms $S,T$ with simple singular spectrum and a set $C$ such that the sequence $ \sum_{n=1}^{N} \mu(S^{ p(n)}C\cap T^{ q(n)}C)/N$ diverges. We give also example of linear non-recurrence for a pair of mixing suspensions of zero entropy and with singular and Lebesgue parts in their spectra.