Spectral theory of regular sequences: parametrisation and spectral characterisation

TL;DR

This paper extends the spectral analysis of regular sequences by introducing ghost measures, parametrising them via compact groups, and classifying their spectral types, including resolving a case of the finiteness conjecture.

Contribution

It generalizes ghost measure existence to broader regular sequences, parametrizes non-unique measures, and links spectral types to matrix spectral properties.

Findings

Ghost measures exist beyond primitive sequences.

Measures can be parametrized by a compact abelian group.

Pure point measures have support in rational numbers.

Abstract

We extend the existence of ghost measures beyond nonnegative primitive regular sequences to a large class of nonnegative real-valued regular sequences. In the general case, where the ghost measure is not unique, we show that they can be parametrised by a compact abelian group. For a subclass of these measures, by replacing primitivity with a commutativity condition, we show that these measures have an infinite convolution structure similar to Bernoulli convolutions. Using this structure, we show that these ghost measures have pure spectral type. Further, we provide results towards a classification of the spectral type based on inequalities involving the spectral radius, joint spectral radius, and Lyapunov exponent of the underlying set of matrices. In the case that the underlying measure is pure point, we show that the support of the measure must be a subset of the rational numbers, a result that resolves a new case of the finiteness conjecture.