Uniformly distributed orbits in ${\mathbb{T}}^d$ and 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 $\ddot{I}‰\mathbf{v}$ to be uniformly distributed modulo $1$ for almost every $\ddot{I}‰\in \mathbb{R}$. We use our results to find new examples of singular substitution $\mathbb{Z}$- and $\mathbb{R}$-actions.

18 pages, accepted version