DOCSLIB.ORG

Springer Monographs in Mathematics for Further Volumes: Tullio Ceccherini-Silberstein R Michel Coornaert

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Springer Monographs in Mathematics for Further Volumes: Tullio Ceccherini-Silberstein R Michel Coornaert Springer Monographs in Mathematics For further volumes: www.springer.com/series/3733 Tullio Ceccherini-Silberstein r Michel Coornaert Cellular Automata and Groups Tullio Ceccherini-Silberstein Michel Coornaert Dipartimento di Ingegneria Institut de Recherche Mathématique Avancée Università del Sannio Université de Strasbourg C.so Garibaldi 107 7 rue René-Descartes 82100 Benevento 67084 Strasbourg Cedex Italy France [email protected] [email protected] ISSN 1439-7382 ISBN 978-3-642-14033-4 e-ISBN 978-3-642-14034-1 DOI 10.1007/978-3-642-14034-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010934641 Mathematics Subject Classification (2010): 37B15, 68Q80, 20F65, 43A07, 16S34, 20C07 © Springer-Verlag Berlin Heidelberg 2010 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To Katiuscia, Giacomo, and Tommaso To Martine and Nathalie Preface Two seemingly unrelated mathematical notions, namely that of an amenable group and that of a cellular automaton, were both introduced by John von Neumann in the first half of the last century. Amenability, which originated from the study of the Banach-Tarski paradox, is a property of groups gener- alizing both commutativity and finiteness. Nowadays, it plays an important role in many areas of mathematics such as representation theory, harmonic analysis, ergodic theory, geometric group theory, probability theory, and dy- namical systems. Von Neumann used cellular automata to serve as theoretical models for self-reproducing machines. About twenty years later, the famous cellular automaton associated with the Game of Life was invented by John Horton Conway and popularized by Martin Gardner. The theory of cellular automata flourished as one of the main branches of computer science. Deep connections with complexity theory and logic emerged from the discovery that some cellular automata are universal Turing machines. AgroupG is said to be amenable (as a discrete group) if the set of all subsets of G admits a right-invariant finitely additive probability measure. All finite groups, all solvable groups (and therefore all abelian groups), and all finitely generated groups of subexponential growth are amenable. Von Neumann observed that the class of amenable groups is closed under the operation of taking subgroups and that the free group of rank two F2 is non- amenable. It follows that a group which contains a subgroup isomorphic to F2 is non-amenable. However, there are examples of groups which are non- amenable and contain no subgroups isomorphic to F2 (the first examples of such groups were discovered by Alexander Y. Ol’shanskii and by Sergei I. Adyan). Loosely speaking, a general cellular automaton can be described as fol- lows. A configuration is a map from a set called the universe into another set called the alphabet. The elements of the universe are called cells and the elements of the alphabet are called states. A cellular automaton is then a map from the set of all configurations into itself satisfying the following local property: the state of the image configuration at a given cell only depends on vii viii Preface the states of the initial configuration on a finite neighborhood of the given cell. In the classical setting, for instance in the cellular automata constructed by von Neumann and the one associated with Conway’s Game of Life, the alphabet is finite, the universe is the two dimensional infinite square lattice, and the neighborhood of a cell consists of the cell itself and its eight adjacent cells. By iterating a cellular automaton one gets a discrete dynamical system. Such dynamical systems have proved very useful to model complex systems arising from natural sciences, in particular physics, biology, chemistry, and population dynamics. ∗ ∗∗ In this book, the universe will always be a group G (in the classical setting the corresponding group was G = Z2) and the alphabet may be finite or infinite. The left multiplication in G induces a natural action of G on the set of configurations which is called the G-shift and all cellular automata will be required to commute with the shift. It was soon realized that the question whether a given cellular automaton is surjective or not needs a special attention. From the dynamical viewpoint, surjectivity means that each configuration may be reached at any time. The first important result in this direction is the celebrated theorem of Moore and Myhill which gives a necessary and sufficient condition for the surjectivity of a cellular automaton with finite alphabet over the group G = Z2.EdwardF. Moore and John R. Myhill proved that such a cellular automaton is surjective if and only if it is pre-injective. As the term suggests it, pre-injectivity is a weaker notion than injectivity. More precisely, a cellular automaton is said to be pre-injective if two configurations are equal whenever they have the same image and coincide outside a finite subset of the group. Moore proved the “surjective ⇒ pre-injective” part and Myhill proved the converse implication shortly after. One often refers to this result as to the Garden of Eden theorem. This biblical terminology is motivated by the fact that, regarding a cellular automaton as a dynamical system with discrete time, a configuration which is not in the image of the cellular automaton may only appear as an initial configuration, that is, at time t =0. The surprising connection between amenability and cellular automata was established in 1997 when Antonio Mach`ı, Fabio Scarabotti and the first au- thor proved the Garden of Eden theorem for cellular automata with finite alphabets over amenable groups. At the same time, and completely inde- pendently, Misha Gromov, using a notion of spacial entropy, presented a more general form of the Garden of Eden theorem where the universe is an amenable graph with a dense holonomy and cellular automata are called maps of bounded propagation. Mach`ı, Scarabotti and the first author also showed that both implications in the Garden of Eden theorem become false if the underlying group contains a subgroup isomorphic to F2. The question whether the Garden of Eden theorem could be extended beyond the class of Preface ix amenable groups remained open until 2008 when Laurent Bartholdi proved that the Moore implication fails to hold for non-amenable groups. As a conse- quence, the whole Garden of Eden theorem only holds for amenable groups. This gives a new characterization of amenable groups in terms of cellular au- tomata. Let us mention that, up to now, the validity of the Myhill implication for non-amenable groups is still an open problem. Following Walter H. Gottschalk, a group G is said to be surjunctive if every injective cellular automaton with finite alphabet over G is surjective. Wayne Lawton proved that all residually finite groups are surjunctive and that ev- ery subgroup of a surjunctive group is surjunctive. Since injectivity implies pre-injectivity, an immediate consequence of the Garden of Eden theorem for amenable groups is that every amenable group is surjunctive. Gromov and Benjamin Weiss introduced a class of groups, called sofic groups, which in- cludes all residually finite groups and all amenable groups, and proved that every sofic group is surjunctive. Sofic groups can be defined in three equivalent ways: in terms of local approximation by finite symmetric groups equipped with their Hamming distance, in terms of local approximation of their Cayley graphs by finite labelled graphs, and, finally, as being the groups that can be embedded into ultraproducts of finite symmetric groups (this last charac- terization is due to G´abor Elek and Endre Szab´o). The class of sofic groups is the largest known class of surjunctive groups. It is not known, up to now, whether all groups are surjunctive (resp. sofic) or not. Stimulated by Gromov ideas, we considered cellular automata whose al- phabets are vector spaces. In this framework, the space of configurations has a natural structure of a vector space and cellular automata are required to be linear. An analogue of the Garden of Eden theorem was proved for linear cellular automata with finite dimensional alphabets over amenable groups. In the proof, the role of entropy, used in the finite alphabet case, is now played by the mean dimension, a notion introduced by Gromov. Also, examples of linear cellular automata with finite dimensional alphabets over groups con- taining F2 showing that the linear version of the Garden of Eden theorem may fail to hold in this case, were provided. It is not known, up to now, if the Garden of Eden theorem for linear cellular automata with finite dimen- sional alphabet only holds for amenable groups or not. We also introduced the notion of linear surjunctivity: a group G is said to be L-surjunctive if ev- ery injective linear cellular automaton with finite dimensional alphabet over G is surjective. We proved that every sofic group is L-surjunctive.
Load more

Recommended publications
  • Surjunctivity and Reversibility of Cellular Automata Over Concrete
    SURJUNCTIVITY AND REVERSIBILITY OF CELLULAR AUTOMATA OVER CONCRETE CATEGORIES TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT Abstract. Following ideas developed by Misha Gromov, we investigate surjunctivity and reversibility properties of cellular automata defined over certain concrete categories. Dedicated to Alessandro Fig`aTalamanca 1. Introduction A cellular automaton is an abstract machine which takes as input a configuration of a universe and produces as output another configuration. The universe consists of cells and a configuration is described by the state of each cell of the universe. There is an input and an output state set and these two sets may be distinct. The state sets are also called the sets of colors, the sets of symbols, or the alphabets. The transition rule of a cellular automaton must obey two important properties, namely locality and time-independence. Locality means that the output state of a given cell only depends on the input states of a finite number of its neighboring cells possibly including the cell itself. In the present paper, we restrict ourselves to cellular automata for which the universe is a group. The cells are the elements of the group. If the input alphabet is denoted by A, the output alphabet by B, and the group by G, this means that a cellular automaton is given by a map τ : AG → BG, where AG := {x: G → A} is the set of all input configurations and BG := {y : G → B} is the set of all possible output configurations. Besides locality, we will also always require a symmetry condition for τ, namely that it commutes with the shift actions of G on AG and BG (see Section 2 for precise definitions).
    [Show full text]
  • Expansive Actions with Specification of Sofic Groups, Strong Topological Markov Property, and Surjunctivity
    EXPANSIVE ACTIONS WITH SPECIFICATION OF SOFIC GROUPS, STRONG TOPOLOGICAL MARKOV PROPERTY, AND SURJUNCTIVITY TULLIO CECCHERINI-SILBERSTEIN, MICHEL COORNAERT, AND HANFENG LI Abstract. A dynamical system is a pair (X; G), where X is a compact metrizable space and G is a countable group acting by homeomorphisms of X. An endomorphism of (X; G) is a continuous selfmap of X which commutes with the action of G. One says that a dynamical system (X; G) is surjunctive provided that every injective endomorphism of (X; G) is surjective (and therefore is a homeomorphism). We show that when G is sofic, every expansive dynamical system (X; G) with nonnegative sofic topological entropy and satisfying the weak specification and the strong topological Markov properties, is surjunctive. Contents 1. Introduction 2 2. Preliminaries 5 2.1. General notation 5 2.2. Actions 5 2.3. Shifts and subshifts 5 2.4. Dynamical systems 6 2.5. Expansivity 6 2.6. The strong topological Markov property 7 2.7. The weak specification property 8 2.8. The pseudo-orbit tracing property 8 2.9. Algebraic dynamical systems 9 2.10. The Hamming distance 9 2.11. Sofic groups 9 2.12. Sofic topological entropy 10 3. Proofs 12 4. Examples 17 Appendix A. Stirling's approximation formula 18 References 19 Date: July 26, 2021. 2010 Mathematics Subject Classification. 37B40, 37B10, 37D20, 20F65. Key words and phrases. Sofic group, sofic entropy, surjunctive dynamical system, expansive action, strong topological Markov property, weak specification property, subshift, surjunctive subshift, strongly irreducible subshift, algebraic dynamical system. 1 2 TULLIO CECCHERINI-SILBERSTEIN, MICHEL COORNAERT, AND HANFENG LI 1.
    [Show full text]
  • Open Problems
    Open Problems In the list below we collect some open problems related to the topics treated in this book. (OP-1) Let G be an amenable periodic group which is not locally finite. Does there exist a finite set A and a cellular automaton τ : AG → AG which is surjective but not injective? (OP-2) Let G be a periodic group which is not locally finite and let A be an infinite set. Does there exist a bijective cellular automaton τ : AG → AG which is not invertible? (OP-3) Let G be a periodic group which is not locally finite and let V be an infinite-dimensional vector space over a field K. Does there exist a bijective linear cellular automaton τ : V G → V G which is not invertible? (OP-4) Is every Gromov-hyperbolic group residually finite (resp. residually amenable, resp. sofic, resp. surjunctive)? (OP-5) (Gottschalk’s conjecture) Is every group surjunctive? (OP-6) Let G be a periodic group which is not locally finite and let A be an infinite set. Does there exist a cellular automaton τ : AG → AG whose image τ(AG) is not closed in AG with respect to the prodiscrete topology? (OP-7) Let G be a periodic group which is not locally finite and let V be an infinite-dimensional vector space over a field K. Does there exist a linear cellular automaton τ : V G → V G whose image τ(V G)isnot closed in V G with respect to the prodiscrete topology? (OP-8) Let G and H be two quasi-isometric groups.
    [Show full text]
  • Arxiv:2008.10565V2 [Math.GR] 26 Oct 2020 N Ftems Motn Lse Fgop Ngoercgopthe Group Geometric T in on Groups Life of a Classes Live Important Sofi Now Most Surjunctive
    ON DUAL SURJUNCTIVITY AND APPLICATIONS MICHAL DOUCHA AND JAKUB GISMATULLIN Abstract. We explore the dual version of Gottschalk’s conjecture recently intro- duced by Capobianco, Kari, and Taati, and the notion of dual surjunctivity in general. We show that dual surjunctive groups satisfy Kaplansky’s direct finiteness conjecture for all fields of positive characteristic. By quantifying the notions of injectivity and post-surjectivity for cellular automata, we show that the image of the full topological shift under an injective cellular automaton is a subshift of finite type in a quantitative way. Moreover we show that dual surjunctive groups are closed under ultraproducts, under elementary equivalence, and under certain semidirect products (using the ideas of Arzhantseva and Gal for the latter); they form a closed subset in the space of marked groups, fully residually dual surjunctive groups are dual surjunctive, etc. We also consider dual surjunctive systems for more general dynamical systems, namely for certain expansive algebraic actions, employing results of Chung and Li. Contents Introduction 1 1. Post-surjectivity and pre-injectivity 3 2. More observations on injectivity and post-surjectivity 5 3. Dual surjunctive groups and ultraproducts 8 4. Direct finiteness conjecture 12 5. Metric direct finiteness conjecture 13 6. Expansive dynamical systems 16 References 19 Introduction In the beginning of the 1970s, W. Gottschalk introduced the following notion. Let G be a group and A a finite set, and let us consider the topological Bernoulli shift G y AG. arXiv:2008.10565v2 [math.GR] 26 Oct 2020 If any injective, continuous, and G-equivariant map T : AG → AG is also surjective, then G is called surjunctive.
    [Show full text]
  • A Koksma–Hlawka Inequality for Simplices
    Springer INdAM Series Volum e 3 Editor-in-Chief V. Ancona Series Editors P. Cannarsa C. Canuto G. Coletti P. Marcellini G. Patrizio T. Ruggeri E. Strickland A. Verra For further volumes: www.springer.com/series/10283 Massimo A. Picardello Editor Trends in Harmonic Analysis Editor Massimo A. Picardello Dipartimento di Matematica Università di Roma “Tor Vergata” Rome, Italy ISSN 2281-518X ISSN 2281-5198 (electronic) Springer INdAM Series ISBN 978-88-470-2852-4 ISBN 978-88-470-2853-1 (eBook) DOI 10.1007/978-88-470-2853-1 Springer Milan Heidelberg New York Dordrecht London Library of Congress Control Number: 2012948292 © Springer-Verlag Italia 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center.
    [Show full text]