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, 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 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.

Bessi, U. (2021). Another point of view on Kusuoka's measure. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 7, 3241-3271 [10.3934/dcds.2020404].

Another point of view on Kusuoka's measure.

Ugo Bessi
2021

Abstract

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, 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 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.
Bessi, U. (2021). Another point of view on Kusuoka's measure. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 7, 3241-3271 [10.3934/dcds.2020404].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/422547
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