On uniform recurrence for hyperbolic automorphisms of the $2$-dimensional torus
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913219656810496 |
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| author | Lamprinakis, Georgios Persson, Tomas Sponheimer, Alejandro Rodriguez |
| author_facet | Lamprinakis, Georgios Persson, Tomas Sponheimer, Alejandro Rodriguez |
| contents | We are interested in studying sets of the form \[ \mathcal{U}(α) :=
\left\{ x\in X: \ \exists M=M(x) \geq 1 \text{ such that } \forall N\geq M, \ \exists n\leq N \text{ such that } d(T^nx, x) \leq |λ|^{-αN} \right\} \] where $(X,T,d)$ is our metric dynamical system and $|λ|>1$. Although a lot of results exist for the one dimensional case, not as many are known for systems in higher dimensions and especially in the hyperbolic case. We consider $X=\mathbb{T}^2$, $T(x) = Ax \pmod{1}$, where $A$ is a hyperbolic, area preserving, $2\times 2$ matrix with integer entries and $λ$ is the eigenvalue of $A$ of modulus larger than $1$ and we explicitly calculate the Hausdorff dimension of this set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00690 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On uniform recurrence for hyperbolic automorphisms of the $2$-dimensional torus Lamprinakis, Georgios Persson, Tomas Sponheimer, Alejandro Rodriguez Dynamical Systems 37D20, 37B20, 28A78 We are interested in studying sets of the form \[ \mathcal{U}(α) := \left\{ x\in X: \ \exists M=M(x) \geq 1 \text{ such that } \forall N\geq M, \ \exists n\leq N \text{ such that } d(T^nx, x) \leq |λ|^{-αN} \right\} \] where $(X,T,d)$ is our metric dynamical system and $|λ|>1$. Although a lot of results exist for the one dimensional case, not as many are known for systems in higher dimensions and especially in the hyperbolic case. We consider $X=\mathbb{T}^2$, $T(x) = Ax \pmod{1}$, where $A$ is a hyperbolic, area preserving, $2\times 2$ matrix with integer entries and $λ$ is the eigenvalue of $A$ of modulus larger than $1$ and we explicitly calculate the Hausdorff dimension of this set. |
| title | On uniform recurrence for hyperbolic automorphisms of the $2$-dimensional torus |
| topic | Dynamical Systems 37D20, 37B20, 28A78 |
| url | https://arxiv.org/abs/2402.00690 |