The Repetition Property for Sequences on Tori Generated by Polynomials or Skew-Shifts

TL;DR

This paper investigates when sequences generated by polynomials or skew-shifts on tori exhibit the repetition property, impacting the spectral analysis of ergodic Schrödinger operators by identifying classes with no eigenvalues.

Contribution

It characterizes the conditions under which such sequences have the repetition property, revealing new classes of ergodic Schrödinger operators without eigenvalues.

Findings

Sequences generated by certain polynomials or skew-shifts have the repetition property.

Some classes of ergodic Schrödinger operators with these potentials have no eigenvalues.

Contradicts earlier beliefs about eigenvalue presence in these systems.

Abstract

The repetition property of a sequence in a metric space, a notion introduced by us in an earlier paper, is of importance in the spectral analysis of ergodic Schr\"odinger operators. It may be used to exclude eigenvalues for such operators. In this paper we study the question of when a sequence on a torus that is generated by a polynomial or a skew-shift has the repetition property. This provides classes of ergodic Schr\"odinger operators with potentials generated by skew-shifts on tori that have, contrary to earlier belief, no eigenvalues.