Boshernitzan's condition, factor complexity, and an application

TL;DR

This paper explores Boshernitzan's condition in minimal subshifts, demonstrating its flexibility by constructing examples with faster-than-expected complexity growth, and applies this to spectral properties of Schrödinger operators.

Contribution

It shows how to construct minimal subshifts satisfying Boshernitzan's condition with superexponential complexity growth, expanding understanding of the condition's restrictiveness.

Findings

Constructed minimal subshifts with faster-than-expected complexity growth satisfying Boshernitzan's condition.

Proved that no subexponentially growing sequence bounds the spectral gaps of associated Schrödinger operators.

Linked complexity growth rates to spectral properties of discrete Schrödinger operators.

Abstract

Boshernitzan found a decay condition on the measure of cylinder sets that implies unique ergodicity for minimal subshifts. Interest in the properties of subshifts satisfying this condition has grown recently, due to a connection with the study of discrete Schr\"odinger operators. Of particular interest is the question of how restrictive Boshernitzan's condition is. While it implies zero topological entropy, our main theorem shows how to construct minimal subshifts satisfying the condition whose factor complexity grows faster than any pre-assigned subexponential rate. As an application, via a theorem of Damanik and Lenz, we show that there is no subexponentially growing sequence for which the spectra of all discrete Schr\"odinger operators associated with subshifts whose complexity grows faster than the given sequence, have only finitely many gaps.