Title: Julia sets with conformal dimension one

Abstract: Complex dynamics is the study of dynamical systems defined by iterating rational maps on the Riemann sphere. For a post-critically finite rational map f, the Julia set $J_f$ is a fractal defined as the repeller of the dynamical system of f. As a fractal embedded in the Riemann sphere, the Julia set has conformal dimension between 1 and 2. The Julia set has conformal dimension 2 if and only if it is the whole sphere. However, the other extreme case, when conformal dimension=1, contains diverse Julia sets, including Julia sets of post-critically finite polynomials, Newton maps, and critically fixed rational maps. In this talk, we show that a Julia set $J_f$ has conformal dimension one if and only if there is an f-invariant graph that has topological entropy zero. In the spirit of Sullivan’s dictionary, we also compare this result with Carrasco and Mackay's work on Gromov hyperbolic groups whose boundaries have conformal dimension one.