Mathematics Colloquia and Seminars

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