Ahlfors-regular conformal dimension and energies of graph maps

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 π, φ : G 1 Ñ G 0 between finite graphs that satisfies a natural expanding condition. Associated to this combinatorial data, for each q ě 1, is a numerical invariant E q rπ, φs, its asymptotic q-conformal energy. We show that the Ahlfors-regular conformal dimension of J f is contained in the interval where E q " 1.Among other applications, we give two families of quartic rational maps with Ahlforsregular conformal dimension approaching 1 and 2, respectively.

Conformal Assouad dimension as the critical exponent for combinatorial modulus

“…In (1.4) and (1.3) we adopt the convention that inf ∅ = ∞. Ahlfors regular conformal dimension is a well-studied notion in complex dynamics and hyperbolic groups; see for example [BK05,BM,CM,HP09,PT,Par]. These notions of conformal dimensions are variants of the one introduced by Pansu [Pan89] and we refer the reader to [MT10] for more background and applications.…”

Conformal Assoaud dimension as the critical exponent for combinatorial modulus

“…In (1.3) and (1.4) we adopt the convention that inf ∅ = ∞. Ahlfors regular conformal dimension is a well-studied notion in complex dynamics and hyperbolic groups; see for example [BK05,BM,CM,HP09,PT,Par]. These notions of conformal dimensions are slight variant of the one introduced by Pansu [Pan89] and we refer the reader to [MT10] for more background and applications.…”