Construction Of Local Regular Dirichlet Forms On Julia Sets And The Sierpi(?)ski Gaskets

This thesis is concerned with Julia sets and the Sierpi???ski gaskets about the con-struction of local regular Dirichlet forms and heat kernel estimates.This thesis can be divided into two parts.In the first part,we give the construction of Dirichlet form and heat kernel estimates on Julia sets.Here,the Julia sets are given by the closure of the set of all repelling periodic points of f_c?z?=z²+c for c in the main cardioid or the±_k¹-bulbs where k?2.First,we use external ray parametrization to construct a strongly local regular conservative Dirichlet form on Julia set where the topological property of external ray parametrization is widely used.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.In the second part,we construct a self-similar strongly local regular Dirichlet form on the Sierpi???ski gasket using?-convergence.This construction has direct corollaries that the local Dirichlet form has a domain of some Besov space and non-local Dirichlet forms can approximate the local Dirichlet form.There are currently two methods of construction of local regular Dirichlet forms on fractal spaces.The first method is probabilistic method developed by Barlow,Bass and Perkins where Brownian motions are the limits of random walks on approximating graphs or reflected Brownian motions on approximating domains.This method can be applied on pcf sets and non-pcf sets.The work includes Barlow and Perkins on the Sierpi???ski gakset^[1]and Barlow and Bass on the Sierpi???ski carpet^[2,3].The second method is analytic method developed by Kigami where local regular Dirichlet forms are the limits of energies on approximating graphs.This method can be applied only on pcf sets.The work includes Kigami on the Sierpi???ski gasket^[4]and on pcf sets^[5].Here,we give a new analytic method where the main tool is?-convergence of Mosco.This method can be applied on pcf sets and non-pcf sets and is simper than probabilistic method.The second part is the realization of this method on the Sierpi???ski gasket.