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Periodicity and Transitivity for Cellular Automata in Besicovitch Topologies [chapter]

F. Blanchard, J. Cervelle, E. Formenti
2003 Lecture Notes in Computer Science  
We study cellular automata (CA) behavior in Besicovitch topology. We solve an open problem about the existence of transitive CA.  ...  We also prove that every CA (in Besicovitch topology) either has a unique fixed point or a countable set of periodic points.  ...  Transitive cellular automata As already recalled, Besicovitch topology was introduced in order to further study CA chaotic behavior and, in particular, sensitivity to initial conditions.  ... 
doi:10.1007/978-3-540-45138-9_17 fatcat:jbt6tbkpozdu5ea6jmldzhnyo4

Some results about the chaotic behavior of cellular automata

F. Blanchard, J. Cervelle, E. Formenti
2005 Theoretical Computer Science  
We study the behavior of cellular automata (CA for short) in the Cantor, Besicovitch and Weyl topologies. We solve an open problem about the existence of transitive CA in the Besicovitch topology.  ...  in the Cantor topology, that is, it is dense and has a central role in the study of surjectivity and injectivity.  ...  Conclusions This paper further investigates the periodic and chaotic behaviors of cellular automata in different topological settings, namely the Cantor, Besicovitch and Weyl spaces and topologies, keeping  ... 
doi:10.1016/j.tcs.2005.06.038 fatcat:nxxzsmbyrbcdlaoz4n5pxjvj3i

Geometry and Dynamics of the Besicovitch and Weyl Spaces [article]

Ville Salo, Ilkka Törmä
2012 arXiv   pre-print
look like cellular automata, and present a new proof for the nonexistence of transitive cellular automata in the Besicovitch space.  ...  In addition, we study canonical projections into subshifts, characterize the cellular automata that are contracting or isometric in the Besicovitch or Weyl spaces, study continuous functions that locally  ...  In Section 7, we give a new proof for the nonexistence of transitive cellular automata on the Besicovitch space.  ... 
arXiv:1204.5371v1 fatcat:r5gbqdik6ralzf4yelsofij7na

Geometry and Dynamics of the Besicovitch and Weyl Spaces [chapter]

Ville Salo, Ilkka Törmä
2012 Lecture Notes in Computer Science  
look like cellular automata, and present a new proof for the nonexistence of transitive cellular automata in the Besicovitch space.  ...  In addition, we study canonical projections into subshifts, characterize the cellular automata that are contracting or isometric in the Besicovitch or Weyl spaces, study continuous functions that locally  ...  In Section 7, we give a new proof for the nonexistence of transitive cellular automata on the Besicovitch space.  ... 
doi:10.1007/978-3-642-31653-1_42 fatcat:dppg7lvnevfvpfwfzt3bbyea44

Damage Spreading and μ-Sensitivity on Cellular Automata [chapter]

Bruno Martin
2000 Lecture Notes in Computer Science  
We show relations between new notions on cellular automata based on topological and measure-theoretical concepts: almost everywhere sensitivity to initial conditions for Besicovitch pseudo-distance, damage  ...  Through natural examples, we illustrate the links between these formal definitions and Wolfram's empirical classification.  ...  We will begin by the definitions of cellular automata, Besicovitch topology and Bernoulli measure.  ... 
doi:10.1007/3-540-44929-9_18 fatcat:vntvltkmxva67adc5myr7fcaei

Dynamics of Cellular Automata in Non-compact Spaces [chapter]

Enrico Formenti, Petr Kůrka
2009 Encyclopedia of Complexity and Systems Science  
Generic space: the space of configurations for which upper-density and lowerdensity coincide.  ...  Spreading set: a clopen invariant set propagating both to the left and to the right. Toeplitz space: the space of regular quasi-periodic configurations.  ...  Acknowledgments We thank Marcus Pivato and Francois Blanchard for careful reading of the paper and many valuable suggestions.  ... 
doi:10.1007/978-0-387-30440-3_138 fatcat:gvxrl2srbjhuhcziqbbwp4gx4y

Dynamics of Cellular Automata in Non-compact Spaces [chapter]

Enrico Formenti, Petr Kůrka
2012 Computational Complexity  
Generic space: the space of configurations for which upper-density and lowerdensity coincide.  ...  Spreading set: a clopen invariant set propagating both to the left and to the right. Toeplitz space: the space of regular quasi-periodic configurations.  ...  Acknowledgments We thank Marcus Pivato and Francois Blanchard for careful reading of the paper and many valuable suggestions.  ... 
doi:10.1007/978-1-4614-1800-9_59 fatcat:ivwmh6vnfrbwzj6l7lgkhbq75u

Topology Inspired Problems for Cellular Automata, and a Counterexample in Topology

Ville Salo, Ilkka Törmä
2012 Electronic Proceedings in Theoretical Computer Science  
We consider two relatively natural topologizations of the set of all cellular automata on a fixed alphabet.  ...  We then generalize this space, in the sense that every shift-invariant measure on the configuration space induces a pseudometric on cellular automata, and study the properties of these spaces.  ...  In hope of finding other uses for topologizing cellular automata, we introduce two naturally arising topologies for the set of cellular automata.  ... 
doi:10.4204/eptcs.90.5 fatcat:efkb3n4dkzdsbclredjyujeehi

Topological chaos: what may this mean?

François Blanchard
2009 Journal of difference equations and applications (Print)  
We confront existing definitions of chaos with the state of the art in topological dynamics.  ...  Among the main ideas in this article are the distinction between overall chaos and partial chaos, and the fact that some dynamical properties may be considered more chaotic than others. 2000 Mathematics  ...  to the talks and discussing with participants.  ... 
doi:10.1080/10236190802385355 fatcat:ds5drmn4kvgurpaftamwwzemgm

Topological chaos: what may this mean ? [article]

François Blanchard
2008 arXiv   pre-print
We confront existing definitions of chaos with the state of the art in topological dynamics.  ...  Among the main ideas in this article are the distinction between overall chaos and partial chaos, and the fact that some dynamical properties may be considered more chaotic than others.  ...  to the talks and discussing with participants.  ... 
arXiv:0805.0232v1 fatcat:xtec5cby7fcylp2bfacrslao6u

Generalized Besicovitch and Weyl spaces: Topology, patterns, and sliding block codes

Silvio Capobianco
2011 Theoretical Computer Science  
The Besicovitch and Weyl topologies on the space of configurations take a point of view completely different from the usual product topology; as such, the properties of the former are much different from  ...  The one-dimensional case has already been the subject of thorough studies; we extend it to greater dimensions, and also to more general groups.  ...  The author also thanks the anonymous referees for their thorough review and helpful suggestions.  ... 
doi:10.1016/j.tcs.2011.02.020 fatcat:wfoizx4wurbp7bgfh3munmi5ai

A complete metric in the set of mixing transformations

Sergei V Tikhonov
2007 Sbornik. Mathematics  
Initially [1] the conjugacy lemma was stated and proved in the following conditions.  ...  We say that Halmos' conjugacy lemma is held for D if a conjugacy class of every free dynamical system T ∈ D is everywhere dense in D.  ...  Topological conjugacy of cellular automata in various subspaces Jeremias Epperlein TU Dresden, Germany jeremias.epperlein@tu-dresden.de We study topological conjugacy of one-dimensional cellular automata  ... 
doi:10.1070/sm2007v198n04abeh003850 fatcat:4qcnoaxtsnc55mtxy34x444uhu

Mean equicontinuity and mean sensitivity on cellular automata [article]

Luguis de los Santos Baños, Felipe García-Ramos
2020 arXiv   pre-print
We show that a cellular automaton (or shift-endomorphism) on a transitive subshift is either almost equicontinuous or sensitive.  ...  On the other hand, we construct a cellular automaton on a full-shift (hence a transitive subshift) that is neither almost mean equicontinuous nor mean sensitive.  ...  So it is natural to ask if, just like in the transitive topological dynamics case, a similar dichotomy to Kurka's holds for the mean versions on cellular automata (on transitive subshifts).  ... 
arXiv:2003.01311v3 fatcat:fcsmrjmhujetzkvv2bqfozjwfu

A brief history of cellular automata

Palash Sarkar
2000 ACM Computing Surveys  
Cellular automata are simple models of computation which exhibit fascinatingly complex behavior.  ...  Here we trace a history of cellular automata from their beginnings with von Neumann to the present day.  ...  Comments and suggestions from anonymous referees helped in improving the presentation and treatment of several topics.  ... 
doi:10.1145/349194.349202 fatcat:llv74746vnbwxcslsvadfiv4xi

A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications [article]

Kamalika Bhattacharjee, Nazma Naskar, Souvik Roy, Sukanta Das
2018 arXiv   pre-print
Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation.  ...  Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modelling different physical, natural  ...  Acknowledgements The authors gratefully acknowledge the anonymous reviewers for their comments and suggestions, which have helped to improve the quality and readability of the paper.  ... 
arXiv:1607.02291v3 fatcat:nvvcvjlupvdtpdz34kii2ovt5m
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