Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (5): 1399-1414.doi: 10.1016/S0252-9602(17)30081-4

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HEAT KERNEL ESTIMATES ON JULIA SETS

Meng YANG   

  1. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
  • Received:2016-01-08 Revised:2017-04-27 Online:2017-10-25 Published:2017-10-25

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 ±(1)/k-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.

Key words: Julia sets, heat kernel, resistance metric

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