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

Rotem Yaari

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 $ω\mathbf{v}$ to be uniformly distributed modulo $1$ for almost every $ω\in\mathbb{R}$. We use our results to find new examples of singular substitution $\mathbb{Z}$- and $\mathbb{R}$-actions.

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

Abstract

We find sufficient conditions for the singularity of a substitution -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 -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 , we find necessary and sufficient conditions for the orbit of to be uniformly distributed modulo for almost every . We use our results to find new examples of singular substitution - and -actions.

Paper Structure

This paper contains 5 sections, 16 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Uniformly distributed sequences and linear recurrences
  3. The minimal subspace
  4. Applications to substitutions
  5. Examples

Key Result

Proposition 2.1

A sequence $(\mathbf{x}_n)_{n=0}^\infty\subset\mathbb{R}^d$ is u.d. mod $1$ if and only if for every non-zero $\mathbf{h}\in\mathbb{Z}^d$ the sequence of real numbers $(\langle\mathbf{x}_n, \mathbf{h}\rangle)_{n=0}^\infty$ is u.d. mod $1$.

Theorems & Definitions (45)

  • Proposition 2.1: kuipers2012uniform
  • Theorem 2.2: kuipers2012uniform
  • Definition 2.3
  • Theorem 2.4
  • Example 2.5
  • Definition 2.6
  • Theorem 2.7: see everest2003recurrence
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10: van der Poorten van1981some, Evertse evertse1984sums
  • ...and 35 more