Irreducible wavelet representations and ergodic automorphisms on solenoids

TL;DR

This paper investigates the irreducibility of wavelet representations, especially those linked to ergodic shifts on solenoids and the Cantor set, by exploring their connections with transfer operators and refinement equations.

Contribution

It establishes new connections between wavelet irreducibility, ergodic automorphisms, and transfer operators, providing multiple equivalent formulations of the problem.

Findings

Identifies conditions for wavelet representation irreducibility

Links between ergodic shifts and fixed points of transfer operators

Provides multiple equivalent formulations of the irreducibility problem

Abstract

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in \cite{DuJo06b}, and we present several equivalent formulations of the problem.