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A survey on transitivity in discrete time dynamical systems. application to symbolic systems and related languages

Published online by Cambridge University Press:  20 July 2006

Gianpiero Cattaneo ,
Alberto Dennunzio  and
Fabio Farina
Gianpiero Cattaneo
Affiliation:
Dipartimento di Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano–Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italy; cattang@disco.unimib.it; dennunzio@disco.unimib.it; farina@disco.unimib.it
Alberto Dennunzio
Affiliation:
Dipartimento di Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano–Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italy; cattang@disco.unimib.it; dennunzio@disco.unimib.it; farina@disco.unimib.it
Fabio Farina
Affiliation:
Dipartimento di Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano–Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italy; cattang@disco.unimib.it; dennunzio@disco.unimib.it; farina@disco.unimib.it
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Abstract

The main goal of this paper is the investigation of a relevantproperty which appears in the various definition of deterministictopological chaos for discrete time dynamical system:transitivity. Starting from the standard Devaney's notion of topological chaosbased on regularity, transitivity, and sensitivity to the initialconditions, the critique formulated by Knudsen is taken intoaccount in order to exclude periodic chaos from this definition.Transitivity (or some stronger versions of it) turns out to be therelevant condition of chaos and its role is discussed by a surveyof some important results about it with the presentation of somenew results. In particular, we study topological mixing, strong transitivity,and full transitivity. Their applications to symbolic dynamics areinvestigated with respect to the relationships with the associatedlanguages.

Keywords

Transitivitychaossymbolic dynamicsformal languages.

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Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 40 , Issue 2: Alberto Bertoni: Climbing summits , April 2006 , pp. 333 - 352
Copyright
© EDP Sciences, 2006

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References

E. Akin, The general topology of dynamical systems. Graduate Stud. Math. 1, American Mathematical Society, Providence (1993).
Akin, E. and Kolyada, S., Li-Yorke sensitivity. Nonlinearity 16 (2003) 1421–1433. CrossRef
Alseda, L.L., Kolyada, S., Llibre, J. and Snoha, L., Entropy and periodic points for transitive maps. Trans. Amer. Math. Soc. 351 (1999) 1551–1573. CrossRef
Alseda, L.L., Del Rio, M.A. and Rodriguez, J.A., A survey on the relation between transitivity and dense periodicity for graph maps. J. Diff. Equ. Appl. 9 (2003) 281–288. CrossRef
Banks, J., Brooks, J., Cairns, G., Davis, G. and Stacey, P., Devaney's de, Onfinition of chaos. Amer. Math. Montly 99 (1992) 332–334. CrossRef
Blanchard, F. and Hansel, G., Languages and subshifts, Automata on Infinite Words (Berlin), edited by M. Nivat and D. Perrin. Lect. Notes Comput. Sci. 192 (1985) 138–146.
Blanchard, F., Kurka, P. and Maas, A., Topological and measure-theoretic properties of one-dimensional cellular automata. Physica D 103 (1997) 86–99. CrossRef
Blanchard, F. and Tisseur, P., Some properties of cellular automata with equicontinuity points, Ann. Inst. Henri Poincaré. Probab. Statist. 36 (2000) 569–582.
Boyle, M. and Kitchens, B., Periodic points for cellular automata. Indag. Math. 10 (1999) 483–493. CrossRef
Cattaneo, G. and Dennunzio, A., Subshift behavior of cellular automata. topological properties and related languages, Machines, Computations, and Universality, in 4th International Conference, MCU 2004 (Berlin). Lect. Notes Comput. Sci. 3354 (2005) 140–152. CrossRef
Cattaneo, G., Dennunzio, A. and Margara, L., Chaotic subshifts and related languages applications to one-dimensional cellular automata. Fundamenta Informaticae 52 (2002) 39–80.
Cattaneo, G., Dennunzio, A. and Margara, L., Solution of some conjectures about topological properties of linear cellular automata. Theoret. Comput. Sci. 325 (2004) 249–271. CrossRef
Cattaneo, G., Formenti, E. and Margara, L., Topological definitions of deterministic chaos, applications to cellular automata dynamics, in Cellular Automata, a Parallel Model, edited by M. Delorme and J. Mazoyer. Kluwer Academic Pub., Dordrecht. Math. Appl. 460 (1999) 213–259.
Codenotti, B. and Margara, L., Transitive cellular automata are sensitive. Amer. Math. Monthly 103 (1996) 58–62. CrossRef
Crannell, A., The role of transitivity in Devaney's definition of chaos. Amer. Math. Monthly 102 (1995) 768–793. CrossRef
M. Denker, C. Grillenberger and K. Sigmund, Ergodic theory on compact spaces. Lect. Notes Math. 527 (1976).
R.L. Devaney, An introduction to chaotic dynamical systems. Second ed., Addison-Wesley (1989).
Eckmann, J.-P. and Ruelle, D., Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57 (1985) 617–656. CrossRef
Glasner, E. and Weiss, B., Sensitive dependence on initial condition. Nonlinearity 6 (1993) 1067–1075. CrossRef
A. Kameyama, Topological transitivity and strong transitivity. Acta Math. Univ. Comenianae LXXI, 139.
V. Kannan and A. Nagar, Topological transitivity for discrete dynamical systems, in Applicable Mathematics in Golden Age, edited by J.C. Misra. Narosa Pub. (2002).
A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems. Cambridge University Press (1995).
J.L. Kelley, General topology. Springer-Verlag (1975).
Knudsen, C., Chaos without nonperiodicity. Amer. Math. Monthly 101 (1994) 563–565. CrossRef
Kolyada, S., Li-Yorke sensitivity and other concepts of chaos. Ukrainian Mathematical Journal 56 (2004) 1242–1257. CrossRef
Kolyada, S. and Snoha, L., Some aspect of topological transitivity – a survey. Grazer Math. Ber. 334 (1997) 3–35.
P. Kurka, Topological and symbolic dynamics, Cours Spécialisés 11. Société Mathématique de France (2004).
J.P. LaSalle, Stability theory for difference equations. MAA Studies in Math., American Mathematical Society (1976).
D. Lind and B. Marcus, An introduction to symbolic dynamics and coding. Cambidge University Press (1995).
Martinez, S., Hyperbolic dynamical systems with isolated points. Lect. Notes Math. 527 (1983) 47–64.
Parry, W., Intrinsic markov chains. Trans. Amer. Math. Soc. 112 (1964) 55–56. CrossRef
Ruelle, D., Strange attractors. Math. Intell. 2 (1980) 126–137. CrossRef
Bau sen Du, On the nature of chaos, arXiv:math.DS/0602585 v1 (February 2006).
Silverman, S., On maps with dense orbits and the definitions of chaos. Rocky Mountain Jour. Math. 22 (1992) 353–375. CrossRef
Vellekoop, M. and Berglund, R., On intervals, transitivity = chaos. Amer. Math. Monthly 101 (1994) 353–355. CrossRef
P. Walters, An introduction to ergodic theory. Springer, Berlin (1982).
Weiss, B., Topological transitivity and ergodic measures. Math. Syst. Theory 5 (1971) 71–5. CrossRef
S. Wiggins, Global bifurcations and chaos. Springer, Berlin (1988).