Ahlfors-regular conformal dimension, abbreviated by ARconfdim, is the infimum of the Hausdorff dimension in a quasisymmetric class of Ahlfors-regular metric spaces. Being embedded in the sphere, Julia sets of post-critically finite rational maps have ARconfdim between 1 and 2. A Julia set has ARconfdim 2 if and only if it is the whole sphere. In this talk, we discuss the other extreme case, when ARconfdim=1, which contains critically fixed rational maps and post-critically finite polynomials or Newton maps. We show that the Julia set of a post-critically finite hyperbolic rational map f has ARconfdim 1 if and only if there exists an f-invariant graph G containing the post-critical set such that the dynamics restricted to G has topological entropy zero. We also discuss Sullivan's dictionary related to this work. Speaker(s): InSung Park (Indiana University)