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

Valery V. Ryzhikov

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} μ(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.

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

Abstract

We answer the question of Frantzikinakis and Host about the convergence of ergodic -averages and consider a more general case. Let sequences satisfy the property Then there exist automorphisms with simple singular spectrum and a set such that the sequence 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.

Paper Structure

This paper contains 4 sections, 33 equations.

Table of Contents

  1. Introduction
  2. Construction of infinite automorphism ${\bf S}$
  3. Proof of the theorem
  4. Linear non-recurrence