Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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
Banks J., Brooks J., Cairns G., Davis G., Stacey P., On Devaney’s definition of chaos, Amer. Math. Monthly, 1992, 99(4), 332–334
Bernardes N.C. Jr., Darji U.B., Graph theoretic structure of maps of the Cantor space, Adv. Math., 2012, 231(3–4), 1655–1680
Blanchard F., Topological chaos: what does this mean?, J. Difference Equ. Appl., 2009, 15(1), 23–46
Blanchard F., Glasner E., Kolyada S., Maass A., On Li-Yorke pairs, J. Reine Angew. Math., 2002, 547, 51–68
Blanchard F., Huang W., Entropy sets, weakly mixing sets and entropy capacity, Discrete Contin. Dyn. Syst., 2008, 20(2), 275–311
Block L.S., Coppel W.A., Dynamics in One Dimension, Lecture Notes in Math., 1513, Springer, Berlin, 1992
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
Bowen R., Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 1971, 153, 401–414
Brin M., Stuck G., Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002
Bruckner A.M., Ceder J., Chaos in terms of the map x ↦ ω(x, f), Pacific J. Math., 1992, 156(1), 63–96
D’Aniello E., Darji U.B., Chaos among self-maps of the Cantor space, J. Math. Anal. Appl., 2011, 381(2), 781–788
D’Aniello E., Darji U.B., Steele T.H., Ubiquity of odometers in topological dynamical systems, Topology Appl., 2008, 156(2), 240–245
Devaney R.L., An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley Stud. Nonlinearity, Addison-Wesley, Redwood City, 1989
Dinaburg E.I., The relation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR, 1970, 190, 19–22 (in Russian)
Glasner E., Weiss B., Sensitive dependence on initial conditions, Nonlinearity, 1993, 6(6), 1067–1075
Grillenberger C., Constructions of strictly ergodic systems I. Given entropy, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1973, 25(4), 323–334
Hochman M., Genericity in topological dynamics, Ergodic Theory Dynam. Systems, 2008, 28(1), 125–165
Huang W., Ye X., Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology Appl., 2002, 117(3), 259–272
Huang W., Ye X., An explicit scattering, non-weakly mixing example and weak disjointness, Nonlinearity, 2002, 15(3), 849–862
Koscielniak P., On the genericity of chaos, Topology Appl., 2007, 154(9), 1951–1955
Li T.Y., Yorke J.A., Period three implies chaos, Amer. Math. Monthly, 1975, 82(10), 985–992
Mai J., Devaney’s chaos implies existence of s-scrambled sets, Proc. Amer. Math. Soc., 2004, 132(9), 2761–2767
Oxtoby J.C., Measure and Category, Grad. Texts in Math., 2, Springer, New York-Heidelberg-Berlin, 1971
Weiss B., Topological transitivity and ergodic measures, Math. Systems Theory, 1971, 5, 71–75
Yano K., A remark on the topological entropy of homeomorphisms, Invent. Math., 1980, 59(3), 215–220
Ye X., Zang R., On sensitive sets in topological dynamics, Nonlinearity, 2008, 21(7), 1601–1620
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
D’Aniello, E., Steele, T.H. Chaotic behaviour of the map x ↦ ω(x, f). centr.eur.j.math. 12, 584–592 (2014). https://doi.org/10.2478/s11533-013-0360-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-013-0360-3