Ahlfors-regular conformal dimension and energies of graph maps
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| Format: | Preprint |
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2021
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| _version_ | 1866912406476685312 |
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| author | Pilgrim, Kevin M. Thurston, Dylan P. |
| author_facet | Pilgrim, Kevin M. Thurston, Dylan P. |
| contents | For a hyperbolic rational map $f$ with connected Julia set, we give upper and lower bounds on the Ahlfors-regular conformal dimension of its Julia set $J_f$ from a family of energies of associated graph maps. Concretely, the dynamics of $f$ is faithfully encoded by a pair of maps $π, ϕ: G_1 \to G_0$ between finite graphs that satisfies a natural expanding condition. Associated to this combinatorial data, for each $q \geq 1$, is a numerical invariant $\overline{E}^q[π,ϕ]$, its asymptotic $q$-conformal energy. We show that the Ahlfors-regular conformal dimension of $J_f$ is contained in the interval where $\overline{E}^q=1$.
Among other applications, we give two families of quartic rational maps with Ahlfors-regular conformal dimension approaching 1 and 2, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_09041 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Ahlfors-regular conformal dimension and energies of graph maps Pilgrim, Kevin M. Thurston, Dylan P. Dynamical Systems Complex Variables Primary 37F10, Secondary 30L10, 20E08 For a hyperbolic rational map $f$ with connected Julia set, we give upper and lower bounds on the Ahlfors-regular conformal dimension of its Julia set $J_f$ from a family of energies of associated graph maps. Concretely, the dynamics of $f$ is faithfully encoded by a pair of maps $π, ϕ: G_1 \to G_0$ between finite graphs that satisfies a natural expanding condition. Associated to this combinatorial data, for each $q \geq 1$, is a numerical invariant $\overline{E}^q[π,ϕ]$, its asymptotic $q$-conformal energy. We show that the Ahlfors-regular conformal dimension of $J_f$ is contained in the interval where $\overline{E}^q=1$. Among other applications, we give two families of quartic rational maps with Ahlfors-regular conformal dimension approaching 1 and 2, respectively. |
| title | Ahlfors-regular conformal dimension and energies of graph maps |
| topic | Dynamical Systems Complex Variables Primary 37F10, Secondary 30L10, 20E08 |
| url | https://arxiv.org/abs/2112.09041 |