## Markov Property of Kusuoka-Zhou's Dirichlet Forms on Self-Similar Sets

J. Math. Sci. Univ. Tokyo
Vol. 7 (2000), No. 1, Page 27--33.

Kigami, Jun
Markov Property of Kusuoka-Zhou's Dirichlet Forms on Self-Similar Sets
The main purpose of this note is to fill a gap in Kusuoka-Zhou's construction of self-similar Dirichlet forms on self- similar sets. Unfortunately, it is not quite clear whether or not the self-similar closed form $\E$ obtained in the proof of Theorem 6.9 of [KZ] satisfies the Markov property. We will use a kind of fixed point theorem of order preserving additive maps on a cone to prove existence of a self-similar closed form with the Markov property. The fixed point theorem will be introduced in \S 1. It is also applicable to other problems, for example, the existence problem of a harmonic structure on a p.c.f. self-similar set. In \S 2, we will apply the fixed point theorem to show existence of self-similar Dirichlet forms on self-similar sets.