Non-recurrence and divergent $\bf ({p(n)},{q(n)})$-averages for deterministic automorphisms
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929548073893888 |
|---|---|
| author | Ryzhikov, Valery V. |
| author_facet | Ryzhikov, Valery V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11787 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-recurrence and divergent $\bf ({p(n)},{q(n)})$-averages for deterministic automorphisms Ryzhikov, Valery V. Dynamical Systems 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. |
| title | Non-recurrence and divergent $\bf ({p(n)},{q(n)})$-averages for deterministic automorphisms |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2410.11787 |