Uniformly distributed orbits in $\mathbb{T}^d$ and singular substitution dynamical systems

TL;DR

This paper establishes conditions for the spectral singularity of substitution dynamical systems and explores the uniform distribution of orbits in toral endomorphisms, leading to new examples of singular actions.

Contribution

It generalizes previous criteria for spectral singularity and links orbit distribution in toral endomorphisms to the spectral properties of substitution actions.

Findings

Conditions for spectral singularity of substitution actions.

Necessary and sufficient conditions for orbit uniform distribution.

New examples of singular substitution dynamical systems.

Abstract

We find sufficient conditions for the singularity of a substitution $\mathbb{Z}$-action's spectrum, which generalize the conditions given in arXiv:2003.11287, Theorem 2.4, and we also obtain a similar statement for a collection of substitution $\mathbb{R}$-actions, including the self-similar one. To achieve this, we first study the distribution of related toral endomorphism orbits. In particular, given a toral endomorphism and a vector $\mathbf{v}\in\mathbb{Q}^d$, we find necessary and sufficient conditions for the orbit of $\omega\mathbf{v}$ to be uniformly distributed modulo $1$ for almost every $\omega\in\mathbb{R}$. We use our results to find new examples of singular substitution $\mathbb{Z}$- and $\mathbb{R}$-actions.