Monday Analysis Seminar: Closure of the Sierpinski gasket minus the unit interval

2012-1-30 14:45 - 2012-1- 16:15
Faculty of Science Buliding #3 Room 210
Jun Kigami (Kyoto University)
The Sierpinski gasket (S.G. for short) is the invariant set with respect to the three contractions with contraction ration 1/2 whose fixed points are the vertices of a equilateral triangle. It is a typical example of a self-similar set and its Hausdorff dimension (with respect to the Euclidean metric) is log 3/log 2 and a primary playground for the study of analysis and stochastic processes on fractals.
In this talk, I am going to reveal a relation between the Brownian motion on the SG and a random walk on a tree. The key observation is that the SG minus the one line segment of the equilateral triangle (the unit interval) has a structure of a tree. In the course of the talk, I will give an introductory review on analysis on the SG including the construction of Brownian motion, asymptotic behaviors of associated heat kernel and the distribution of the eigenvalues of associated Laplacian.