Almost sure dimensional properties for the spectrum and the density of states of Sturmian Hamiltonians
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912314704265216 |
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| author | Cao, Jie Qu, Yanhui |
| author_facet | Cao, Jie Qu, Yanhui |
| contents | In this paper, we find a full Lebesgue measure set of frequencies $\check \II\subset [0,1]\setminus \Q$ such that for any $(α,λ)\in \check \II\times [24,\infty)$, the Hausdorff and box dimensions of the spectrum of the Sturmian Hamiltonian $H_{α,λ,θ}$ coincide and are independent of $α$. Denote the common value by $D(λ)$, we show that $D(λ)$ satisfies a Bowen type formula, and is locally Lipschitz. We obtain the exact asymptotic behavior of $D(λ)$ as $λ$ tends to $ \infty.$ This considerably improves the result of Damanik and Gorodetski (Comm. Math. Phys. 337, 2015). We also show that for any $(α,λ)\in \check \II\times [24,\infty)$, the density of states measure of $H_{α,λ,θ}$ is exact-dimensional; its Hausdorff and packing dimensions coincide and are independent of $α$. Denote the common value by $d(λ)$, we show that $d(λ)$ satisfies a Young type formula, and is Lipschitz. We obtain the exact asymptotic behavior of $d(λ)$ as $λ$ tends to $ \infty.$ During the course of study, we also answer several questions in the same paper of Damanik and Gorodetski. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07305 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Almost sure dimensional properties for the spectrum and the density of states of Sturmian Hamiltonians Cao, Jie Qu, Yanhui Spectral Theory Dynamical Systems 37A30 In this paper, we find a full Lebesgue measure set of frequencies $\check \II\subset [0,1]\setminus \Q$ such that for any $(α,λ)\in \check \II\times [24,\infty)$, the Hausdorff and box dimensions of the spectrum of the Sturmian Hamiltonian $H_{α,λ,θ}$ coincide and are independent of $α$. Denote the common value by $D(λ)$, we show that $D(λ)$ satisfies a Bowen type formula, and is locally Lipschitz. We obtain the exact asymptotic behavior of $D(λ)$ as $λ$ tends to $ \infty.$ This considerably improves the result of Damanik and Gorodetski (Comm. Math. Phys. 337, 2015). We also show that for any $(α,λ)\in \check \II\times [24,\infty)$, the density of states measure of $H_{α,λ,θ}$ is exact-dimensional; its Hausdorff and packing dimensions coincide and are independent of $α$. Denote the common value by $d(λ)$, we show that $d(λ)$ satisfies a Young type formula, and is Lipschitz. We obtain the exact asymptotic behavior of $d(λ)$ as $λ$ tends to $ \infty.$ During the course of study, we also answer several questions in the same paper of Damanik and Gorodetski. |
| title | Almost sure dimensional properties for the spectrum and the density of states of Sturmian Hamiltonians |
| topic | Spectral Theory Dynamical Systems 37A30 |
| url | https://arxiv.org/abs/2310.07305 |