Invariant measures for subshifts arising from substitutions of some primitive components

Hama, Masaki; Yuasa, Hisatoshi

doi:10.48550/arxiv.1003.3364

Cited by 1 publication

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“…Also, in the quest of exploring Schrödinger operators over symbolic dynamical systems that exhibit an intermediate (combinatorial and topological) complexity, we may want to consider further generalizations of almost primitive substitutions. One such class of examples, called "substitutions of some primitive components", was studied in [23]. Our interest in substitutions over infinite alphabets was mostly directed towards structural properties of a very specific class of examples of constant length.…”

Section: Discussionmentioning

confidence: 99%

Spectral properties of Schrödinger operators associated to almost minimal substitution systems

Eichinger,

Gohlke

2020

Preprint

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We study the spectral properties of ergodic Schrödinger operators that are associated to 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. 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 topologically conjugate to an odometer.

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Section: Discussionmentioning

confidence: 99%