A Note on the Harnack Inequality Related with the Martin Boundary

#### H.-J. Ruan , X.-Y. Wang

2015, v.21, №2, 283-292

ABSTRACT

Harnack inequality plays a key role for the homeomorphism between the Martin boundaries and the self-similar sets.
In this note, we study a transition probability on the finite words space of an iterated function system, whose invariant set is the unit interval. We present the necessary and sufficient condition for the transition probability
to satisfy the Harnack inequality.

Keywords: harmonic functions, iterated function system, Martin boundary, self-similar sets, open set condition, transition probability