Uniform quasi-multiplicativity of locally constant cocycles and applications
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866917579918934016 |
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| author | Mohammadpour, Reza Park, Kiho |
| author_facet | Mohammadpour, Reza Park, Kiho |
| contents | In this paper, we show that a locally constant cocycle $\mathcal{A}$ is $k$-quasi multiplicative under the irreducibility assumption. More precisely, we show that if $\mathcal{A}^t$ and $\mathcal{A}^{\wedge m}$ are irreducible for every $t \mid d$ and $1\leq m \leq d-1$, then $\mathcal{A}$ is $k$-uniformly spannable for some $k\in \mathbb{N}$, which implies that $\mathcal{A}$ is $k$-quasi multiplicative. We apply our results to show that the unique subadditive equilibrium Gibbs state is $ψ$-mixing and calculate the Hausdorff dimension of cylindrical shrinking target and recurrence sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_08999 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Uniform quasi-multiplicativity of locally constant cocycles and applications Mohammadpour, Reza Park, Kiho Dynamical Systems Number Theory 28A80, 37A45, 37D35, 37H15 In this paper, we show that a locally constant cocycle $\mathcal{A}$ is $k$-quasi multiplicative under the irreducibility assumption. More precisely, we show that if $\mathcal{A}^t$ and $\mathcal{A}^{\wedge m}$ are irreducible for every $t \mid d$ and $1\leq m \leq d-1$, then $\mathcal{A}$ is $k$-uniformly spannable for some $k\in \mathbb{N}$, which implies that $\mathcal{A}$ is $k$-quasi multiplicative. We apply our results to show that the unique subadditive equilibrium Gibbs state is $ψ$-mixing and calculate the Hausdorff dimension of cylindrical shrinking target and recurrence sets. |
| title | Uniform quasi-multiplicativity of locally constant cocycles and applications |
| topic | Dynamical Systems Number Theory 28A80, 37A45, 37D35, 37H15 |
| url | https://arxiv.org/abs/2209.08999 |