Density of spectral gap property for positively expansive dynamics and smooth potentials, with applications to the phase transition problem
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918038927835136 |
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| author | Bomfim, Thiago Carneiro, Victor |
| author_facet | Bomfim, Thiago Carneiro, Victor |
| contents | It is known that all uniformly expanding dynamics $f: M \rightarrow M$ have no phase transition with respect to a Hölder continuous potential $ϕ: M \rightarrow \mathbb{R}$, in other words, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f , tϕ)$ is analytical. Moreover, the associated transfer operator $\mathcal{L}_{f , tϕ}$, acting on the space of Hölder continuous functions, has the spectral gap property for $t \in \mathbb{R}$. For dynamics that are topologically conjugate to an expanding map, a full understanding has yet to be achieved. On the one hand, by \cite{KQW21,KQ22}, for such maps and continuous potentials, the associated topological pressure function can behave wildly. On the other hand, by \cite{BF23}, for transitive local diffeomorphisms on the circle and a large class of Hölder continuous potentials, the phase transition does not occur, and the associated transfer operator has the spectral gap property for all parameters $t \in \mathbb{R}$. As a first approach to understanding what happens in high dimensions, in this paper, we study positively expansive local diffeomorphisms. In particular, we show that the associated transfer operator has the spectral gap property for a large class of regular potentials. Moreover, for a class of intermittent skew-products and a large class of regular potentials, we obtain phase transition results analogous to \cite{BF23}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_23934 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Density of spectral gap property for positively expansive dynamics and smooth potentials, with applications to the phase transition problem Bomfim, Thiago Carneiro, Victor Dynamical Systems 82B26, 37D35, 37C30, 37C40 It is known that all uniformly expanding dynamics $f: M \rightarrow M$ have no phase transition with respect to a Hölder continuous potential $ϕ: M \rightarrow \mathbb{R}$, in other words, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f , tϕ)$ is analytical. Moreover, the associated transfer operator $\mathcal{L}_{f , tϕ}$, acting on the space of Hölder continuous functions, has the spectral gap property for $t \in \mathbb{R}$. For dynamics that are topologically conjugate to an expanding map, a full understanding has yet to be achieved. On the one hand, by \cite{KQW21,KQ22}, for such maps and continuous potentials, the associated topological pressure function can behave wildly. On the other hand, by \cite{BF23}, for transitive local diffeomorphisms on the circle and a large class of Hölder continuous potentials, the phase transition does not occur, and the associated transfer operator has the spectral gap property for all parameters $t \in \mathbb{R}$. As a first approach to understanding what happens in high dimensions, in this paper, we study positively expansive local diffeomorphisms. In particular, we show that the associated transfer operator has the spectral gap property for a large class of regular potentials. Moreover, for a class of intermittent skew-products and a large class of regular potentials, we obtain phase transition results analogous to \cite{BF23}. |
| title | Density of spectral gap property for positively expansive dynamics and smooth potentials, with applications to the phase transition problem |
| topic | Dynamical Systems 82B26, 37D35, 37C30, 37C40 |
| url | https://arxiv.org/abs/2505.23934 |