Substitution Dynamical Systems-Spectral Analysis
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1294
Martine Queffelec
Substitution Dynamical SystemsSpectral Analysis
SpringerVerlag Berlin Heidelberg New York London Paris Tokyo
Author
Martine Oueffelec Universite Paris XIII, Departernent de Mathematiques 93430 Villetaneuse, France
Mathematics Subject Classification (1980): 11 K28, D43 ISBN 3-540-18692-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18692-1 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
The subject of this book is the spectral analysis of substitution dynamical systems.
i have tried to bring the study up to date, and in view to deserve a large audience, I have specially detailed the background, giving sometimes elementary rather than short proofs. Substitution dynamical systems might be considered as a source of examples in ergodic theory and spectral theory. We present a rather different approach, consisting, given some substitution dynamical system, in a deep study (when possible) of its spectrum which means spectrum of the unitary operator adjoint to this system. Many problems still subsist, in particular for substitution of nonconstant length. I have tried to write this areount in English but I am aware of that I have not perfectly succeeded. I beg the indulgence of the persevering reader. Most of topics tackled in this book have been discussed in our "groupe de travail", mainly with
B. Host, J.F. Mt§la and F. Parreau. It is my
gn:atpleasure to thank them for their very efficient help and assistance. During the period of the work on my dissertation, I benefited from the visit of many specialists in the subject. For numerous useful conversations, I express my gratitude to M. Dekking and M. Keane who initiated us
to
substitutions, and also S. Ferenczi, T. Kamae,
J. Lacroix, M. Lemanczyk, P. Liardet, M. Mendes France, J. Peyriere, G. Rauzy, J.M. Strelcyn, J.P. Thouvenot ••. I have received many encouragements to write this book ; I hope their originators are not too much disappointed now T. Ramsey reread the first part of the manuscript and made useful remarks. I thank him very much, and I do not forget C. Simon for her typing.
Paris,
Juillet 1987.
INTRODUCTION
Our purpose is a complete and unified description of the spectrum of dynamical systems arising from substitutions of constant length (under mild hypotheses). The very attractive feature of this analysis
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