Regular sequences and the joint spectral radius

TL;DR

This paper introduces a growth exponent for k-regular sequences and demonstrates its equivalence to the joint spectral radius of certain matrices derived from the sequence's k-kernel, enabling classification of their growth.

Contribution

It establishes a novel connection between the growth of k-regular sequences and the joint spectral radius of associated matrices, providing a new classification tool.

Findings

The growth exponent equals the joint spectral radius for k-regular sequences.

Provides a method to classify sequence growth based on matrix spectral properties.

Introduces the notion of a growth exponent for k-regular sequences.

Abstract

We classify the growth of a $k$-regular sequence based on information from its $k$-kernel. In order to provide such a classification, we introduce the notion of a growth exponent for $k$-regular sequences and show that this exponent is equal to the joint spectral radius of any set of a special class of matrices determined by the $k$-kernel.