Julia sets with Ahlfors-regular conformal dimension one

2022/09/27 by Insung Park, Park, Insung
Mathematics · #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2209.13384

openalex publication_date 2022/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a post-critically finite hyperbolic rational map f, we show that its Julia set 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.

Related

• Ahlfors-regular conformal dimension and energies of graph maps
• Invariant graphs in Julia sets and decompositions of rational maps
• A canonical decomposition of postcritically finite rational maps and their maximal expanding quotients
• Hausdorff dimension and conformal measures of Feigenbaum Julia sets
• Decomposition of rational maps by stable multicurves