Polynomial approximation of self-similar measures and the spectrum of the transfer operator

TL;DR

This paper investigates the spectral properties of transfer operators associated with self-similar measures, providing polynomial eigenfunctions and approximations, with applications to Bernoulli convolutions.

Contribution

It identifies polynomial eigenfunctions of the transfer operator and introduces polynomial approximations of self-similar measures, advancing understanding of their spectral structure.

Findings

Eigenvalues of the Hutchinson operator determined

Polynomial eigenfunctions of the transfer operator identified

Natural polynomial approximations of self-similar measures developed

Abstract

We consider self-similar measures on $\mathbb R.$ The Hutchinson operator $H$ acts on measures and is the dual of the transfer operator $T$ which acts on continuous functions. We determine polynomial eigenfunctions of $T .$ As a consequence, we obtain eigenvalues of $H$ and natural polynomial approximations of the self-similar measure. Bernoulli convolutions are studied as an example.