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 dynamics of f. 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 whole sphere. However, the other extreme case, when conformal dimension=1, contains diverse Julia sets, including all Julia sets of post-critically finite polynomials or Newton 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 recent work on Gromov hyperbolic groups.