Abstract:

The conformal dimension of a metric space is a quasisymmetric numerical invariant, which was introduced by P.Pansu in order to classify, up to quasi-isometry, homogeneous spaces of negative curvature. It provides an invariant under topological conjugation in the context of complex dynamical systems. In this talk we give sufficient conditions for the Julia set of a rational map to have conformal dimension one. These conditions are satisfied for (semi)hyperbolic polynomials with connected Julia set. The main ingredient of the proof is a combinatorial version of the conformal modulus from complex analysis.