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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2208.03807 |
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| _version_ | 1866910434141929472 |
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| author | Calta, Kariane Kraaikamp, Cor Schmidt, Thomas A. |
| author_facet | Calta, Kariane Kraaikamp, Cor Schmidt, Thomas A. |
| contents | We give two results for deducing dynamical properties of piecewise Möbius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both follow from mild finiteness conditions on the planar extension along with a new property ``bounded non-full range" used to relax traditional Markov conditions. Second, the ``quilting" operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise Möbius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada $α$-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_03807 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Proofs of ergodicity of piecewise Möbius interval maps using planar extensions Calta, Kariane Kraaikamp, Cor Schmidt, Thomas A. Dynamical Systems 37A25 (Primary) 11K50, 37A10, 37A44 (Secondary) We give two results for deducing dynamical properties of piecewise Möbius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both follow from mild finiteness conditions on the planar extension along with a new property ``bounded non-full range" used to relax traditional Markov conditions. Second, the ``quilting" operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise Möbius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada $α$-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups. |
| title | Proofs of ergodicity of piecewise Möbius interval maps using planar extensions |
| topic | Dynamical Systems 37A25 (Primary) 11K50, 37A10, 37A44 (Secondary) |
| url | https://arxiv.org/abs/2208.03807 |