Transfer operators and dimension of bad sets for non-uniform Fuchsian lattices
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866914396440100864 |
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| author | Marchese, Luca |
| author_facet | Marchese, Luca |
| contents | The set of real numbers which are badly approximable by rationals admits an exhaustion by sets Bad($ε$), whose dimension converges to 1 as $ε$ goes to zero. D. Hensley computed the asymptotic for the dimension up to the first order in $ε$, via an analogous estimate for the set of real numbers whose continued fraction has all entries uniformly bounded. We consider diophantine approximations by parabolic fixed points of any non-uniform lattice in PSL(2,R) and a geometric notion of $ε$-badly approximable points. We compute the dimension of the set of such points up to the first order in $ε$, via the thermodynamic method of Ruelle and Bowen. Geometric good approximations are related to a notion of bounded partial quotients for the Bowen-Series expansion. This gives a family of Cantor sets and associated quasi-compact transfer operators, with simple and positive maximal eigenvalue. Perturbative analysis of spectra applies. Our techniques only apply to non-uniform lattices admitting a finite index free subgroup satisfying a specific property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_11921 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Transfer operators and dimension of bad sets for non-uniform Fuchsian lattices Marchese, Luca Dynamical Systems Number Theory The set of real numbers which are badly approximable by rationals admits an exhaustion by sets Bad($ε$), whose dimension converges to 1 as $ε$ goes to zero. D. Hensley computed the asymptotic for the dimension up to the first order in $ε$, via an analogous estimate for the set of real numbers whose continued fraction has all entries uniformly bounded. We consider diophantine approximations by parabolic fixed points of any non-uniform lattice in PSL(2,R) and a geometric notion of $ε$-badly approximable points. We compute the dimension of the set of such points up to the first order in $ε$, via the thermodynamic method of Ruelle and Bowen. Geometric good approximations are related to a notion of bounded partial quotients for the Bowen-Series expansion. This gives a family of Cantor sets and associated quasi-compact transfer operators, with simple and positive maximal eigenvalue. Perturbative analysis of spectra applies. Our techniques only apply to non-uniform lattices admitting a finite index free subgroup satisfying a specific property. |
| title | Transfer operators and dimension of bad sets for non-uniform Fuchsian lattices |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/1812.11921 |