Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.16939 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915034055049216 |
|---|---|
| author | Villamil, Christian Camilo Silva |
| author_facet | Villamil, Christian Camilo Silva |
| contents | Let $φ$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $Λ$ be a mixing horseshoe of $φ$. Given a smooth real function $f$ defined on $S$, we define for points $η$ in the unstable Cantor set of the pair $(φ,Λ)$, a generalization, $k_{φ,Λ,f}(η)$, of the best constant of Diophantine approximation for irrational numbers. We study the set of points $η$ for which the sets $k_{φ,Λ,f}^{-1}((-\infty,η])$ and $k_{φ,Λ,f}^{-1}(η)$ have the same Hausdorff dimension and when the Hausdorff dimension of $Λ$ is less than one, we describe generically the local Hausdorff dimension of the dynamical Lagrange spectrum, $\mathcal{L}_{φ,Λ,f}$, restricted to this set of points. Finally, we recover the same results for the classical Lagrange spectra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16939 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concentration of dimension in the Lagrange spectrum Villamil, Christian Camilo Silva Dynamical Systems Number Theory Let $φ$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $Λ$ be a mixing horseshoe of $φ$. Given a smooth real function $f$ defined on $S$, we define for points $η$ in the unstable Cantor set of the pair $(φ,Λ)$, a generalization, $k_{φ,Λ,f}(η)$, of the best constant of Diophantine approximation for irrational numbers. We study the set of points $η$ for which the sets $k_{φ,Λ,f}^{-1}((-\infty,η])$ and $k_{φ,Λ,f}^{-1}(η)$ have the same Hausdorff dimension and when the Hausdorff dimension of $Λ$ is less than one, we describe generically the local Hausdorff dimension of the dynamical Lagrange spectrum, $\mathcal{L}_{φ,Λ,f}$, restricted to this set of points. Finally, we recover the same results for the classical Lagrange spectra. |
| title | Concentration of dimension in the Lagrange spectrum |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2411.16939 |