Ergodic Theory of Kusuoka Measures

TL;DR

This paper studies the Kusuoka measure on self-similar fractals using ergodic theory, revealing its spectral properties and decay of correlations, and extends results to generalized Sierpiński gaskets.

Contribution

It demonstrates that the transfer operator for the Kusuoka measure has a spectral gap, leading to exponential decay of correlations and generalizes Bernoulli measures to higher dimensions.

Findings

Spectral gap for the transfer operator on a Banach space containing Hölder functions.

Exponential decay of correlations for the Kusuoka measure.

Explicit convergence rates for generalized Sierpiński gaskets.

Abstract

In the analysis on self-similar fractal sets, the Kusuoka measure plays an important role (cf. \cite{kusuoka2}, \cite{kajino}, \cite{str3}). Here we investigate the Kusuoka measure from an ergodic theoretic viewpoint, seen as an invariant measure on a symbolic space. Our investigation shows that the Kusuoka measure generalizes Bernoulli measures and their properties to higher dimensions of an underlying finite dimensional vector space. Our main result is that the transfer operator on functions has a spectral gap when restricted to a certain Banach space that contains the H\"older continuous functions, as well as the highly discontinuous $g$-function associated to the Kusuoka measure. As a consequence, we obtain exponential decay of correlations. In addition, we provide some explicit rates of convergence for a family of generalized Sierpi\'ski gaskets.