A note on spectral properties of random $S$-adic systems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916909495091200 |
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| author | Solomyak, Boris |
| author_facet | Solomyak, Boris |
| contents | The paper is concerned with random $S$-adic systems arising from an i.i.d. sequence of unimodular substitutions. Using equidistribution results of Benoist and Quint, we show in Theorem 3.3 that, under some natural assumptions, if the Lyapunov exponent of the spectral cocycle is strictly less that 1/2 of the Lyapunov exponent of the random walk on $SL(2,\mathbb{R})$ driven by the sequence of substitution matrices, then almost surely the spectrum of the $S$-adic $\mathbb{Z}$-action is singular with respect to any (fixed in advance) continuous measure. Finally, the appendix (by Pascal Hubert and Carlos Matheus) discusses the weak-mixing property for random $S$-adic systems associated to the family of substitutions introduced in Example 4.1. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_08884 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on spectral properties of random $S$-adic systems Solomyak, Boris Dynamical Systems 37A30, 37B10 The paper is concerned with random $S$-adic systems arising from an i.i.d. sequence of unimodular substitutions. Using equidistribution results of Benoist and Quint, we show in Theorem 3.3 that, under some natural assumptions, if the Lyapunov exponent of the spectral cocycle is strictly less that 1/2 of the Lyapunov exponent of the random walk on $SL(2,\mathbb{R})$ driven by the sequence of substitution matrices, then almost surely the spectrum of the $S$-adic $\mathbb{Z}$-action is singular with respect to any (fixed in advance) continuous measure. Finally, the appendix (by Pascal Hubert and Carlos Matheus) discusses the weak-mixing property for random $S$-adic systems associated to the family of substitutions introduced in Example 4.1. |
| title | A note on spectral properties of random $S$-adic systems |
| topic | Dynamical Systems 37A30, 37B10 |
| url | https://arxiv.org/abs/2403.08884 |