Probability measures on solenoids corresponding to fractal wavelets

TL;DR

This paper investigates measures on generalized solenoids related to fractal wavelets, revealing fundamental differences from classical cases, particularly regarding atomicity of fiber measures and implications for wavelet systems.

Contribution

It demonstrates that measures on fractal-based solenoids cannot be atomic, contrasting with classical filters, and explores the implications for wavelet systems and representations.

Findings

Classical solenoid measures produce atomic fiber measures.

Fractal solenoid measures are non-atomic.

Implications for wavelet systems and group representations.

Abstract

The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen is analyzed further by writing the solenoid as the product of a torus and a Cantor set. Using this decomposition, key differences are revealed between solenoid measures associated with classical filters in $\mathbb R^d$ and those associated with filters on inflated fractal sets. In particular, it is shown that the classical case produces atomic fiber measures, and as a result supports both suitably defined solenoid MSF wavelets and systems of imprimitivity for the corresponding wavelet representation of the generalized Baumslag-Solitar group. In contrast, the fiber measures for filters on inflated fractal spaces cannot be atomic, and thus can support neither MSF wavelets nor systems of imprimitivity.