Phase transition of capacity for the uniform $G_δ$-sets
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866917079167270912 |
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| author | Kleptsyn, Victor Quintino, Fernando |
| author_facet | Kleptsyn, Victor Quintino, Fernando |
| contents | We consider a family of dense $G_δ$ subsets of $[0,1]$, defined as intersections of unions of small uniformly distributed intervals, and study their capacity. Changing the speed at which the lengths of generating intervals decrease, we observe a sharp phase transition from full to zero capacity. Such a $G_δ$ set can be considered as a toy model for the set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products.
Our re-distribution construction can be considered as a generalization of a method applied by Ursell in his construction of a counter-example to a conjecture by Nevanlinna. Also, we propose a simple Cauchy-Schwartz inequality-based proof of related theorems by Lindeberg and by Erdös and Gillis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1910_07653 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Phase transition of capacity for the uniform $G_δ$-sets Kleptsyn, Victor Quintino, Fernando Dynamical Systems 31A15, 31C15, 28A12 We consider a family of dense $G_δ$ subsets of $[0,1]$, defined as intersections of unions of small uniformly distributed intervals, and study their capacity. Changing the speed at which the lengths of generating intervals decrease, we observe a sharp phase transition from full to zero capacity. Such a $G_δ$ set can be considered as a toy model for the set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products. Our re-distribution construction can be considered as a generalization of a method applied by Ursell in his construction of a counter-example to a conjecture by Nevanlinna. Also, we propose a simple Cauchy-Schwartz inequality-based proof of related theorems by Lindeberg and by Erdös and Gillis. |
| title | Phase transition of capacity for the uniform $G_δ$-sets |
| topic | Dynamical Systems 31A15, 31C15, 28A12 |
| url | https://arxiv.org/abs/1910.07653 |