Pure discrete spectrum for a class of one-dimensional substitution tiling systems

Marcy Barge

doi:10.3934/dcds.2016.36.1159

Article Contents

2016, Volume 36, Issue 3: 1159-1173. Doi: 10.3934/dcds.2016.36.1159

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Pure discrete spectrum for a class of one-dimensional substitution tiling systems

Marcy Barge^1,

1.
Department of Mathematics, Montana State University, Bozeman, MT 59717

Received: October 31, 2014

Revised: May 31, 2015

Published: May 2017

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Abstract

We prove that if a primitive and non-periodic substitution is injective on initial letters, constant on final letters, and has Pisot inflation, then the $\mathbb{R}$-action on the corresponding tiling space has pure discrete spectrum. As a consequence, all $\beta$-substitutions for $\beta$ a Pisot simple Parry number have tiling dynamical systems with pure discrete spectrum, as do the Pisot systems arising, for example, from substitutions associated with the Jacobi-Perron and Brun continued fraction algorithms.

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Mathematics Subject Classification: Primary: 37B05, 37B50; Secondary: 11A55, 11A63.

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