Abstract
We study the spectral properties of ergodic Schrödinger operators that are associated with a certain family of non-primitive substitutions on a binary alphabet. The corresponding subshifts provide examples of dynamical systems that go beyond minimality, unique ergodicity and linear complexity. In some parameter region, we are naturally in the setting of an infinite ergodic measure. The almost sure spectrum is singular and contains an interval. We show that under certain conditions, eigenvalues can appear. Some criteria for the exclusion of eigenvalues are fully characterized, including the existence of strongly palindromic sequences. Many of our structural insights rely on return word decompositions in the context of non-uniformly recurrent sequences. We introduce an associated induced system that is conjugate to an odometer.
Highlights
We are interested in a family of discrete Schrodinger operators Hw, where w ∈ X and (X, S, μ) is an ergodic symbolic dynamical system
We introduce the return word substitutionand study the structure of the corresponding Toeplitz sequences
This gives rise to an ergodic measure on X which is infinite precisely if every primitive letter has vanishing densities
Summary
We are interested in a family of discrete Schrodinger operators Hw, where w ∈ X and (X, S, μ) is an ergodic symbolic dynamical system. Vol 22 (2021) Spectral Properties of Schrodinger Operators minimal and uniquely ergodic dynamical system This case has been considered in [13,14], where it was shown that the spectral properties (1)–(3), mentioned for primitive substitutions still hold, and eigenvalues can be excluded using similar techniques. The superscript “ep” stands for eventually periodic, which characterizes the sequences in Xep. All of the properties in Theorem 1.1 follow from structural results on X , most of which can already be found in [37]. As mentioned when we discussed primitive substitutions, an important strategy to exclude eigenvalues almost surely is to show that almost every w ∈ X gives rise to a Gordon potential. 4, we restrict to the two-alphabet case and show that the corresponding family of substitutions is still general enough to yield a variety of different complexity functions.
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