Martin boundary and exit space on the Sierpinski gasket. (English) Zbl 1241.31014
Summary: We define a new Markov chain on the symbolic space representing the Sierpinski gasket (SG), and show that the corresponding Martin boundary is homeomorphic to the SG while the minimal Martin boundary is the three vertices of the SG. In addition, the harmonic structure induced by the Markov chain coincides with the canonical one on the SG. This suggests another approach to consider the existence of Laplacians on those self-similar sets for which the problem is still not settled.
MSC:
| 31C35 | Martin boundary theory |
| 60J50 | Boundary theory for Markov processes |
| 28A80 | Fractals |
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
Keywords:
Martin boundary; Markov chain; Green function; harmonic function; fractal; Sierpinski gasketReferences:
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