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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.07819 |
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| _version_ | 1866908672148373504 |
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| author | Lima, Davi Moreira, Carlos Gustavo Villamil, Christian Camilo Silva |
| author_facet | Lima, Davi Moreira, Carlos Gustavo Villamil, Christian Camilo Silva |
| contents | Let $φ_0$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a transitive horseshoe of $φ_0$. Given a smooth real function $f$ defined in $S$ and a small smooth conservative perturbation $φ$ of $φ_0$, let $L_{φ, f}$ and $M_{φ, f}$ be respectively the Lagrange and Markov spectra associated to the hyperbolic continuation $Λ(φ)$ of the horseshoe $Λ_0$ and $f$. We show that for generic choices of $φ$ and $f$, the Hausdorff dimension of the sets $L_{φ, f}\cap (-\infty, t)$ and $M_{φ, f}\cap (-\infty, t)$ are equal and determine a continuous function as $t\in \mathbb{R}$ varies; generalizing then the Cerqueira-Matheus-Moreira theorem to horseshoes with arbitrary Hausdorff dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07819 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continuity of fractal dimensions in conservative generic Markov and Lagrange dynamical spectra Lima, Davi Moreira, Carlos Gustavo Villamil, Christian Camilo Silva Dynamical Systems Let $φ_0$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a transitive horseshoe of $φ_0$. Given a smooth real function $f$ defined in $S$ and a small smooth conservative perturbation $φ$ of $φ_0$, let $L_{φ, f}$ and $M_{φ, f}$ be respectively the Lagrange and Markov spectra associated to the hyperbolic continuation $Λ(φ)$ of the horseshoe $Λ_0$ and $f$. We show that for generic choices of $φ$ and $f$, the Hausdorff dimension of the sets $L_{φ, f}\cap (-\infty, t)$ and $M_{φ, f}\cap (-\infty, t)$ are equal and determine a continuous function as $t\in \mathbb{R}$ varies; generalizing then the Cerqueira-Matheus-Moreira theorem to horseshoes with arbitrary Hausdorff dimension. |
| title | Continuity of fractal dimensions in conservative generic Markov and Lagrange dynamical spectra |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2305.07819 |