Constructing Laplacians on Limit Spaces of Self-similar Groups

Abstract

This paper is basically an explanatory survey focusing on a new and stimulating connection between Analysis on fractals and self-similar group Theory. It is shown how to construct Dirichlet forms on fractals which are obtained as limit spaces of contracting self-similar groups acting on regular rooted trees. The key idea is to give a prefractal approximation of the limit space by using the sequence of finite Schreier graphs of the action of the group: under certain conditions of compatibility, a Dirichlet form on the limit space is obtained as limit of the sequence of finite Dirichlet forms associated with the classical discrete Laplacian on them. Some known and new examples are described.KeywordsDirichlet formLaplacianPost-critically finite self-similar fractalLimit spaceJulia setSchreier graphSelf-similar groupMathematics Subject Classification (2010)28A8031C2520E0837F50