Uniformly distributed orbits in $\mathbb{T}^d$ and singular substitution dynamical systems
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866917773967360000 |
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| author | Yaari, Rotem |
| author_facet | Yaari, Rotem |
| contents | We find sufficient conditions for the singularity of a substitution $\mathbb{Z}$-action's spectrum, which generalize the conditions given in arXiv:2003.11287, Theorem 2.4, and we also obtain a similar statement for a collection of substitution $\mathbb{R}$-actions, including the self-similar one. To achieve this, we first study the distribution of related toral endomorphism orbits. In particular, given a toral endomorphism and a vector $\mathbf{v}\in\mathbb{Q}^d$, we find necessary and sufficient conditions for the orbit of $ω\mathbf{v}$ to be uniformly distributed modulo $1$ for almost every $ω\in\mathbb{R}$. We use our results to find new examples of singular substitution $\mathbb{Z}$- and $\mathbb{R}$-actions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_13882 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Uniformly distributed orbits in $\mathbb{T}^d$ and singular substitution dynamical systems Yaari, Rotem Dynamical Systems We find sufficient conditions for the singularity of a substitution $\mathbb{Z}$-action's spectrum, which generalize the conditions given in arXiv:2003.11287, Theorem 2.4, and we also obtain a similar statement for a collection of substitution $\mathbb{R}$-actions, including the self-similar one. To achieve this, we first study the distribution of related toral endomorphism orbits. In particular, given a toral endomorphism and a vector $\mathbf{v}\in\mathbb{Q}^d$, we find necessary and sufficient conditions for the orbit of $ω\mathbf{v}$ to be uniformly distributed modulo $1$ for almost every $ω\in\mathbb{R}$. We use our results to find new examples of singular substitution $\mathbb{Z}$- and $\mathbb{R}$-actions. |
| title | Uniformly distributed orbits in $\mathbb{T}^d$ and singular substitution dynamical systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2108.13882 |