Atsushi Imai
Pentakun, the mod 5 Markov chain and a Martin boundary
Preprint series: Mathematica Gottingensis

MSC:

60J10 Markov chains with discrete parameter

31C05 Harmonic, subharmonic, superharmonic functions

60J50 Boundary theory

Abstract: Let ${\mathcal P}$ denote the p.c.f. self-similar set defined by mapping the regular pentagon into itself by five similarities each leaving one vertex fixed.
We define the canonical Markov chain for ${\mathcal P}$ and denote its Markov operator by $P$.
We show that its Martin boundary ${\mathcal M}$ is homeomorphic to ${\mathcal P}$.
The associated Dirichlet problem $(P-I)f=0$ and $f=g$ on ${\mathcal P}$ has a unique solution such that $f(\xi)={\mathcal P}_{\xi}$ for $\xi \in {\mathcal P}$.
We obtain an integral representation for kernel functions on ${\mathcal P}$ (Poisson integral type).
Keywords: Martin boundary, Markov chain