Ahlfors-regular conformal dimension and energies of graph maps

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 $\pi ,\varphi :G_1\to G_0$ between finite graphs that satisfies a natural expanding condition. Associated to this combinatorial data, for each $q\ge 1$, is a numerical invariant ${\bar{E}}^q[\pi ,\varphi ]$, its asymptotic $q$-conformal energy. We show that the Ahlfors-regular conformal dimension of $J_f$ is contained in the interval where ${\bar{E}}^q=1$. Among other applications, we give two families of quartic rational maps with Ahlfors-regular conformal dimension approaching 1 and 2, respectively.