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

2024/10/15 by Ryzhikov, Valery V.
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.11787

Abstract

We answer the question of Frantzikinakis and Host about the convergence of ergodic (n²,n³)-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) → +∞. Then there exist automorphisms S,T with simple singular spectrum and a set C such that the sequence ∑_n=1^N μ(S ^p(n)C∩ 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.

Related

• Non-commuting transformations with non-converging 2-fold ergodic averages
• Spectra and joint dynamics of Poisson suspensions for rank-one automorphisms
• Rates of Divergence of non-Conventional Ergodic Averages
• Strictly ergodic models and the convergence of non-conventional pointwise ergodic averages
• On the pointwise convergence of multiple ergodic averages