# American Institute of Mathematical Sciences

July  2021, 41(7): 3241-3271. doi: 10.3934/dcds.2020404

## Another point of view on Kusuoka's measure

 Dipartimento di Matematica e Fisica, Largo S. Leonardo Murialdo 1, 00146, Roma

Received  December 2019 Revised  September 2020 Published  July 2021 Early access  December 2020

Fund Project: Work partially supported by the PRIN2009 grant "Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations"

Kusuoka's measure on fractals is a Gibbs measure of a very special kind, since its potential is discontinuous while the standard theory of Gibbs measures requires continuous (in its simplest version, Hölder) 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 bilinear forms on fractals. Moreover, we shall see that Kusuoka's measure and bilinear form can be recovered in a simple way from the matrix-valued Gibbs measure.

Citation: Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404
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##### References:
The first stage of the harmonic gasket
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