Ahlfors-regular conformal dimension and energies of graph maps

TL;DR

This paper establishes bounds on the Ahlfors-regular conformal dimension of Julia sets of hyperbolic rational maps using energies derived from associated graph maps, linking dynamical properties to combinatorial invariants.

Contribution

It introduces a novel approach to estimate conformal dimensions via energies of graph maps, providing explicit bounds and examples for specific rational maps.

Findings

Conformal dimension bounds are linked to graph map energies.

Explicit examples of rational maps with conformal dimensions approaching 1 and 2.

Abstract

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 $\pi, \phi : G_1 \to G_0$ between finite graphs that satisfies a natural expanding condition. Associated to this combinatorial data, for each $q \geq 1$, is a numerical invariant $\overline{E}^q[\pi,\phi]$, 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.