Julia sets with Ahlfors-regular conformal dimension one
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910060298371072 |
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| author | Park, Insung |
| author_facet | Park, Insung |
| contents | For a post-critically finite hyperbolic rational map $f$, we show that its Julia set $\mathcal{J}_f$ has Ahlfors-regular conformal dimension one if and only if $f$ is a crochet map, i.e., there is an $f$-invariant connected graph $G$ containing the post-critical set such that $f|_G$ has topological entropy zero. We use finite subdivision rules to obtain graph virtual endomorphisms, which are 1-dimensional models of post-critically finite rational maps, and we approximate the asymptotic conformal energies of graph virtual endomorphisms to estimate the Ahlfors-regular conformal dimensions of Julia sets. To prove the main theorem, we also establish the monotonicity of asymptotic conformal energies under the decomposition of rational maps by invariant multicurves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_13384 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Julia sets with Ahlfors-regular conformal dimension one Park, Insung Dynamical Systems Complex Variables Metric Geometry 37F10 For a post-critically finite hyperbolic rational map $f$, we show that its Julia set $\mathcal{J}_f$ has Ahlfors-regular conformal dimension one if and only if $f$ is a crochet map, i.e., there is an $f$-invariant connected graph $G$ containing the post-critical set such that $f|_G$ has topological entropy zero. We use finite subdivision rules to obtain graph virtual endomorphisms, which are 1-dimensional models of post-critically finite rational maps, and we approximate the asymptotic conformal energies of graph virtual endomorphisms to estimate the Ahlfors-regular conformal dimensions of Julia sets. To prove the main theorem, we also establish the monotonicity of asymptotic conformal energies under the decomposition of rational maps by invariant multicurves. |
| title | Julia sets with Ahlfors-regular conformal dimension one |
| topic | Dynamical Systems Complex Variables Metric Geometry 37F10 |
| url | https://arxiv.org/abs/2209.13384 |