Minimal covers with continuity-preserving transfer operators for topological dynamical systems
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| Format: | Preprint |
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2024
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| _version_ | 1866914920577105920 |
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| author | Brix, Kevin Aguyar Hume, Jeremy B. Li, Xin |
| author_facet | Brix, Kevin Aguyar Hume, Jeremy B. Li, Xin |
| contents | Given a non-invertible dynamical system with a transfer operator, we show there is a minimal cover with a transfer operator that preserves continuous functions. We also introduce an essential cover with even stronger continuity properties. For one-sided sofic subshifts, this generalizes the Krieger and Fischer covers, respectively. Our construction is functorial in the sense that certain equivariant maps between dynamical systems lift to equivariant maps between their covers, and these maps also satisfy better regularity properties. As applications, we identify finiteness conditions which ensure that the thermodynamic formalism is valid for the covers. This establishes the thermodynamic formalism for a large class of non-invertible dynamical systems, e.g. certain piecewise invertible maps. When applied to semi-étale groupoids, our minimal covers produce étale groupoids which are models for $C^*$-algebras constructed by Thomsen. The dynamical covers and groupoid covers are unified under the common framework of topological graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_11917 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal covers with continuity-preserving transfer operators for topological dynamical systems Brix, Kevin Aguyar Hume, Jeremy B. Li, Xin Dynamical Systems Operator Algebras Primary 37C30, Secondary 22A22, 37A55 Given a non-invertible dynamical system with a transfer operator, we show there is a minimal cover with a transfer operator that preserves continuous functions. We also introduce an essential cover with even stronger continuity properties. For one-sided sofic subshifts, this generalizes the Krieger and Fischer covers, respectively. Our construction is functorial in the sense that certain equivariant maps between dynamical systems lift to equivariant maps between their covers, and these maps also satisfy better regularity properties. As applications, we identify finiteness conditions which ensure that the thermodynamic formalism is valid for the covers. This establishes the thermodynamic formalism for a large class of non-invertible dynamical systems, e.g. certain piecewise invertible maps. When applied to semi-étale groupoids, our minimal covers produce étale groupoids which are models for $C^*$-algebras constructed by Thomsen. The dynamical covers and groupoid covers are unified under the common framework of topological graphs. |
| title | Minimal covers with continuity-preserving transfer operators for topological dynamical systems |
| topic | Dynamical Systems Operator Algebras Primary 37C30, Secondary 22A22, 37A55 |
| url | https://arxiv.org/abs/2408.11917 |