Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników

Czasopismo

2014 | 12 | 4 | 584-592

Tytuł artykułu

Chaotic behaviour of the map x ↦ ω(x, f)

Treść / Zawartość

Warianty tytułu

Języki publikacji

EN

Abstrakty

EN
Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ω f and some forms of chaos are investigated.

Słowa kluczowe

Wydawca

Czasopismo

Rocznik

Tom

12

Numer

4

Strony

584-592

Opis fizyczny

Daty

wydano
2014-04-01
online
2014-01-17

Twórcy

  • Seconda Università degli Studi di Napoli
  • Weber State University

Bibliografia

  • [1] Akin E., Auslander J., Berg K., When is a transitive map chaotic?, In: Convergence in Ergodic Theory and Probability, Columbus, June, 1993, Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996
  • [2] Banks J., Brooks J., Cairns G., Davis G., Stacey P., On Devaney’s definition of chaos, Amer. Math. Monthly, 1992, 99(4), 332–334 http://dx.doi.org/10.2307/2324899
  • [3] Bernardes N.C. Jr., Darji U.B., Graph theoretic structure of maps of the Cantor space, Adv. Math., 2012, 231(3–4), 1655–1680 http://dx.doi.org/10.1016/j.aim.2012.05.024
  • [4] Blanchard F., Topological chaos: what does this mean?, J. Difference Equ. Appl., 2009, 15(1), 23–46 http://dx.doi.org/10.1080/10236190802385355
  • [5] Blanchard F., Glasner E., Kolyada S., Maass A., On Li-Yorke pairs, J. Reine Angew. Math., 2002, 547, 51–68
  • [6] Blanchard F., Huang W., Entropy sets, weakly mixing sets and entropy capacity, Discrete Contin. Dyn. Syst., 2008, 20(2), 275–311
  • [7] Block L.S., Coppel W.A., Dynamics in One Dimension, Lecture Notes in Math., 1513, Springer, Berlin, 1992
  • [8] Blokh A., Bruckner A.M., Humke P.D., Smítal J., The space of ω-limit sets of a continuous map of the interval, Trans. Amer. Math. Soc., 1996, 348(4), 1357–1372 http://dx.doi.org/10.1090/S0002-9947-96-01600-5
  • [9] Bowen R., Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 1971, 153, 401–414 http://dx.doi.org/10.1090/S0002-9947-1971-0274707-X
  • [10] Brin M., Stuck G., Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002 http://dx.doi.org/10.1017/CBO9780511755316
  • [11] Bruckner A.M., Ceder J., Chaos in terms of the map x ↦ ω(x, f), Pacific J. Math., 1992, 156(1), 63–96 http://dx.doi.org/10.2140/pjm.1992.156.63
  • [12] D’Aniello E., Darji U.B., Chaos among self-maps of the Cantor space, J. Math. Anal. Appl., 2011, 381(2), 781–788 http://dx.doi.org/10.1016/j.jmaa.2011.03.065
  • [13] D’Aniello E., Darji U.B., Steele T.H., Ubiquity of odometers in topological dynamical systems, Topology Appl., 2008, 156(2), 240–245 http://dx.doi.org/10.1016/j.topol.2008.07.003
  • [14] Devaney R.L., An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley Stud. Nonlinearity, Addison-Wesley, Redwood City, 1989
  • [15] Dinaburg E.I., The relation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR, 1970, 190, 19–22 (in Russian)
  • [16] Glasner E., Weiss B., Sensitive dependence on initial conditions, Nonlinearity, 1993, 6(6), 1067–1075 http://dx.doi.org/10.1088/0951-7715/6/6/014
  • [17] Grillenberger C., Constructions of strictly ergodic systems I. Given entropy, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1973, 25(4), 323–334 http://dx.doi.org/10.1007/BF00537161
  • [18] Hochman M., Genericity in topological dynamics, Ergodic Theory Dynam. Systems, 2008, 28(1), 125–165 http://dx.doi.org/10.1017/S0143385707000521
  • [19] Huang W., Ye X., Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology Appl., 2002, 117(3), 259–272 http://dx.doi.org/10.1016/S0166-8641(01)00025-6
  • [20] Huang W., Ye X., An explicit scattering, non-weakly mixing example and weak disjointness, Nonlinearity, 2002, 15(3), 849–862 http://dx.doi.org/10.1088/0951-7715/15/3/320
  • [21] Koscielniak P., On the genericity of chaos, Topology Appl., 2007, 154(9), 1951–1955 http://dx.doi.org/10.1016/j.topol.2007.01.014
  • [22] Li T.Y., Yorke J.A., Period three implies chaos, Amer. Math. Monthly, 1975, 82(10), 985–992 http://dx.doi.org/10.2307/2318254
  • [23] Mai J., Devaney’s chaos implies existence of s-scrambled sets, Proc. Amer. Math. Soc., 2004, 132(9), 2761–2767 http://dx.doi.org/10.1090/S0002-9939-04-07514-8
  • [24] Oxtoby J.C., Measure and Category, Grad. Texts in Math., 2, Springer, New York-Heidelberg-Berlin, 1971 http://dx.doi.org/10.1007/978-1-4615-9964-7
  • [25] Weiss B., Topological transitivity and ergodic measures, Math. Systems Theory, 1971, 5, 71–75 http://dx.doi.org/10.1007/BF01691469
  • [26] Yano K., A remark on the topological entropy of homeomorphisms, Invent. Math., 1980, 59(3), 215–220 http://dx.doi.org/10.1007/BF01453235
  • [27] Ye X., Zang R., On sensitive sets in topological dynamics, Nonlinearity, 2008, 21(7), 1601–1620 http://dx.doi.org/10.1088/0951-7715/21/7/012

Typ dokumentu

Bibliografia

Identyfikatory

Identyfikator YADDA

bwmeta1.element.doi-10_2478_s11533-013-0360-3
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.