Topics In Mathematical Physics On Sierpinski Carpets

Abstract

We study three topics in mathematical physics on fractal domains which are based on the Sierpinski carpet and its higher-dimensional analogs. First, we rigorously investigate the thermodynamics of the ideal massive and massless Bose gas, from which quantitative results about Bose-Einstein condensation, blackbody radiation, and the (zero- and finitetemperature) Casimir effect are obtained. Second, we prove the subsequential Mosco convergence of discrete Dirichlet forms on Sierpinski carpet graphs, and from there deduce the convergence of the discrete Green forms. Last but not least, we enumerate a collection of periodic billiard orbits in a planar self-similar Sierpinski carpet billiard table, which paves the way for future studies of billiard dynamics on fractal billiards.