Curve attractors for marked rational maps
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910311183810560 |
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| author | Smith, Zachary |
| author_facet | Smith, Zachary |
| contents | A Thurston map $f\colon (S^2, A) \to (S^2, A)$ with marking set $A$ induces a pullback relation on isotopy classes of Jordan curves in $(S^2, A)$. If every curve lands in a finite list of possible curve classes after iterating this pullback relation, then the pair $(f,A)$ is said to have a finite global curve attractor. It is conjectured by Pilgrim that all rational Thurston maps that are not flexible Lattès maps have a finite global curve attractor. We present partial progress on this problem. Specifically, we prove that if $A$ has four points and the postcritical set (which is a subset of $A$) has two or three points, then $(f,A)$ has a finite global curve attractor.
We also discuss extensions of the main result to certain special cases where $f$ has four postcritical points and $A=P_f$. Additionally, we speculate on how some of these ideas might be used in the more general case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curve attractors for marked rational maps Smith, Zachary Dynamical Systems 37F20 (Primary) 37F10, 37F34 (Secondary) A Thurston map $f\colon (S^2, A) \to (S^2, A)$ with marking set $A$ induces a pullback relation on isotopy classes of Jordan curves in $(S^2, A)$. If every curve lands in a finite list of possible curve classes after iterating this pullback relation, then the pair $(f,A)$ is said to have a finite global curve attractor. It is conjectured by Pilgrim that all rational Thurston maps that are not flexible Lattès maps have a finite global curve attractor. We present partial progress on this problem. Specifically, we prove that if $A$ has four points and the postcritical set (which is a subset of $A$) has two or three points, then $(f,A)$ has a finite global curve attractor. We also discuss extensions of the main result to certain special cases where $f$ has four postcritical points and $A=P_f$. Additionally, we speculate on how some of these ideas might be used in the more general case. |
| title | Curve attractors for marked rational maps |
| topic | Dynamical Systems 37F20 (Primary) 37F10, 37F34 (Secondary) |
| url | https://arxiv.org/abs/2401.16636 |