Multivariate Frequent Stability and Diam-Mean Equicontinuity
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909454991097856 |
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| author | Haupt, Lino |
| author_facet | Haupt, Lino |
| contents | In this paper, we introduce and investigate multivariate versions of frequent stability and diam-mean equicontinuity. Given a natural number $m > 1$, we call those notions "frequent $m$-stability" and "diam-mean $m$-equicontinuity". We use these dynamical rigidity properties to characterise systems whose factor map to the maximal equicontinuous factor (MEF) is finite-to-one for a residual set, called "almost finite-to-one extensions", or a set of full measure, called "almost surely finite-to-one extensions". In the case of a $σ$-compact, locally compact, abelian acting group it is shown that frequently $(m+1)$-stable systems are equivalently characterised as almost $m$-to-one extensions of their MEF. Similarly, it is shown that a system is diam-mean $(m+1)$-equicontinuous if and only if it is an almost surely $m$-to-one extension of its MEF. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07038 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multivariate Frequent Stability and Diam-Mean Equicontinuity Haupt, Lino Dynamical Systems 37B05 (Primary) 37A05 37A15 37B02 (Secondary) In this paper, we introduce and investigate multivariate versions of frequent stability and diam-mean equicontinuity. Given a natural number $m > 1$, we call those notions "frequent $m$-stability" and "diam-mean $m$-equicontinuity". We use these dynamical rigidity properties to characterise systems whose factor map to the maximal equicontinuous factor (MEF) is finite-to-one for a residual set, called "almost finite-to-one extensions", or a set of full measure, called "almost surely finite-to-one extensions". In the case of a $σ$-compact, locally compact, abelian acting group it is shown that frequently $(m+1)$-stable systems are equivalently characterised as almost $m$-to-one extensions of their MEF. Similarly, it is shown that a system is diam-mean $(m+1)$-equicontinuous if and only if it is an almost surely $m$-to-one extension of its MEF. |
| title | Multivariate Frequent Stability and Diam-Mean Equicontinuity |
| topic | Dynamical Systems 37B05 (Primary) 37A05 37A15 37B02 (Secondary) |
| url | https://arxiv.org/abs/2501.07038 |