Cyclic Renormalization and Automorphism Groups of Rooted Trees
The theme of the monograph is an interplay between dynamical systems and group theory. The authors formalize and study "cyclic renormalization", a phenomenon which appears naturally for some interval dynamical systems. A possibly infinite hierarchy of suc
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1621
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1621
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Hyman Bass, Maria Victoria Otero-Espinar Daniel Rockmore, Charles Tresser
Cyclic Renormalization and Automorphism Groups of Rooted Trees
Springer
Authors Hyman Bass Dept. of Mathematics Columbia Uiversity NY, NY 10027, USA E-mail: [email protected]
Daniel Rockmore Dept. of Mathematics Dartmouth College Hanover, NH 03755, USA E-mail: [email protected]
Maria Victoria Otero-Espinar Facultad de Matematicas Campus Universitario 15706 Santiago de Compostela Spain
Charles Tresser IBM, T. J. Watson Research Center Yorktown Heights, NY 10598, USA E-mail: tresserwwatson.fbm.com
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Cyclic renormallzation and automorphism groups of rooted trees / Hyman Bass ... - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris Tokyo: Springer, 1995 (Lecture notes in mathematics; 1621) ISBN 3-540-60595-9 NE: Bass, Hyman; GT
Mathematics Subject Classification (1991): 20B25, 20B27, 20E22, 20E08, 20F38, 54H20, 58F03, 58F08 ISBN 3-540-60595-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10479691 46/3142-543210 - Printed on acid-free paper
I am the Lorax, I speak for the trees... t
For the ones we love ...
t From The Lorax, Dr. Seuss
Table of Contents o.
Introduction
I.
Cyclic Renormalization O. 1. 2. 3. 4.
5. 6.
II.
XI 1
Introduction . Renormalization. . . . . . . . Appendix A: Denjoy expansion Interval renormalization . . .. Systems with prescribed renormalizations Infinite interval renormalizability . . . . . Appendix B: Embedding ordered Cantor-like sets in real intervals . . . . . . . . . . . . . . . . . Interval renormalization and periodic points. Self-similarity operators. . .
Itinerary Calculus and Renormalization O. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Motivation
26 35
42 46 47
53 53
Introduction . Preliminaries. . . . . . . . . . . . . . Appendix C: The multi-modal case .. Maximal elements; the quadratic case Maximal elements; the non-quadratic case The *-product . . . . . The *-product theorem . Shift dynamics on Go U GoC . The *-product renormalization theorem Iterated *-products
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