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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.18940 |
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| _version_ | 1866918284579831808 |
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| author | Villamil, Christian Camilo Silva |
| author_facet | Villamil, Christian Camilo Silva |
| contents | Let $φ_0$ be a $C^2$-conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a mixing horseshoe of $φ_0$. Given a smooth real function $f$ defined in $S$ and some diffeomorphism $φ$, close to $φ_0$, let $\mathcal{L}_{φ, f}$ be the Lagrange spectrum associated to the hyperbolic continuation $Λ(φ)$ of the horseshoe $Λ_0$ and $f$. We show that, for generic choices of $φ$ and $f$, if $L_{φ, f}$ is the map that gives the Hausdorff dimension of the set $\mathcal{L}_{φ, f}\cap (-\infty, t)$ for $t\in \mathbb{R}$, then there are at most two points that can be limit of a infinite sequence of discontinuities of $L_{φ, f}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18940 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrum Villamil, Christian Camilo Silva Dynamical Systems Let $φ_0$ be a $C^2$-conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a mixing horseshoe of $φ_0$. Given a smooth real function $f$ defined in $S$ and some diffeomorphism $φ$, close to $φ_0$, let $\mathcal{L}_{φ, f}$ be the Lagrange spectrum associated to the hyperbolic continuation $Λ(φ)$ of the horseshoe $Λ_0$ and $f$. We show that, for generic choices of $φ$ and $f$, if $L_{φ, f}$ is the map that gives the Hausdorff dimension of the set $\mathcal{L}_{φ, f}\cap (-\infty, t)$ for $t\in \mathbb{R}$, then there are at most two points that can be limit of a infinite sequence of discontinuities of $L_{φ, f}$. |
| title | On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrum |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2403.18940 |