Self-similar fractals as boundaries of networks

Details Self-similar fractals as boundaries of networks Self-similar fractals as boundaries of networks

2011-04-08

arxiv.org/abs/1104.1650v1

Mathematics

Probability

25 pages, 5 figures

Scientific paper

For a given pcf self-similar fractal, a certain network (weighted graph) is constructed whose ideal boundary is (homeomorphic to) the fractal. This construction is the first representation of a connected self-similar fractal as the boundary of a reversible Markov chain (i.e., a simple random walk on a network). The boundary construction is effected using certain functions of finite energy which behave like bump functions on the boundary. The random walk is shown to converge to the boundary almost surely, with respect to the standard measure on its trajectory space.

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