Eliminating Thurston obstructions and controlling dynamics on curves
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909394985287680 |
|---|---|
| author | Bonk, Mario Hlushchanka, Mikhail Iseli, Annina |
| author_facet | Bonk, Mario Hlushchanka, Mikhail Iseli, Annina |
| contents | Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$-sphere $S^2$ induces a pull-back operation on Jordan curves $α\subset S^2\setminus P_f$, where $P_f$ is the postcritical set of $f$. Here the isotopy class $[f^{-1}(α)]$ (relative to $P_f$) only depends on the isotopy class $[α]$. We study this operation for Thurston maps with four postcritical points. In this case a Thurston obstruction for the map $f$ can be seen as a fixed point of the pull-back operation.
We show that if a Thurston map $f$ with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can "blow up" suitable arcs in the underlying $2$-sphere and construct a new Thurston map $\widehat f$ for which this obstruction is eliminated. We prove that no other obstruction arises and so $\widehat f$ is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points.
We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_06938 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Eliminating Thurston obstructions and controlling dynamics on curves Bonk, Mario Hlushchanka, Mikhail Iseli, Annina Dynamical Systems Complex Variables 37F20, 37F10 Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$-sphere $S^2$ induces a pull-back operation on Jordan curves $α\subset S^2\setminus P_f$, where $P_f$ is the postcritical set of $f$. Here the isotopy class $[f^{-1}(α)]$ (relative to $P_f$) only depends on the isotopy class $[α]$. We study this operation for Thurston maps with four postcritical points. In this case a Thurston obstruction for the map $f$ can be seen as a fixed point of the pull-back operation. We show that if a Thurston map $f$ with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can "blow up" suitable arcs in the underlying $2$-sphere and construct a new Thurston map $\widehat f$ for which this obstruction is eliminated. We prove that no other obstruction arises and so $\widehat f$ is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem. |
| title | Eliminating Thurston obstructions and controlling dynamics on curves |
| topic | Dynamical Systems Complex Variables 37F20, 37F10 |
| url | https://arxiv.org/abs/2105.06938 |