Heat kernel estimates on Julia sets

Abstract

We give heat kernel estimates on Julia sets J(fc) for quadratic polynomials fc(z)= z2 + c for c in the main cardioid or the ±1k-bulbs where k ≥2. First we use external ray parametrization to construct a regular, strongly local and conservative Dirichlet form on Julia set. Then we show that this Dirichlet form is a resistance form and the corresponding resistance metric induces the same topology as Euclidean metric. Finally, we give heat kernel estimates under the resistance metric.