Another point of view on Kusuoka's measure

TL;DR

This paper explores a new perspective on Kusuoka's measure by constructing matrix-valued Gibbs measures on fractals, enabling the use of standard Gibbs theory and facilitating the development of self-similar Dirichlet forms.

Contribution

It introduces a class of matrix-valued Gibbs measures applicable to fractals, extending standard Gibbs theory to discontinuous potentials and connecting to Kusuoka's measure.

Findings

Matrix-valued Gibbs measures can be constructed within standard Gibbs theory.

These measures can be used to build self-similar Dirichlet forms on fractals.

Kusuoka's measure can be derived from the matrix-valued Gibbs measures.

Abstract

Kusuoka's measure on fractals is a Gibbs measure of a very special kind, because its potential is discontinuous, while the standard theory of Gibbs measures requires continuous (actuallly, H\"older) potentials. In this paper, we shall see that for many fractals it is possible to build a class of matrix-valued Gibbs measures completely within the scope of the standard theory; there are naturally some minor modifications, but they are only due to the fact that we are dealing with matrix-valued functions and measures. We shall use these matrix-valued Gibbs measures to build self-similar Dirichlet forms on fractals. Moreover, we shall see that Kusuoka's measure can be recovered in a simple way from the matrix-valued Gibbs measure.