Topological horseshoes
HTML articles powered by AMS MathViewer
- by Judy Kennedy and James A. Yorke PDF
- Trans. Amer. Math. Soc. 353 (2001), 2513-2530 Request permission
Abstract:
When does a continuous map have chaotic dynamics in a set $Q$? More specifically, when does it factor over a shift on $M$ symbols? This paper is an attempt to clarify some of the issues when there is no hyperbolicity assumed. We find that the key is to define a “crossing number” for that set $Q$. If that number is $M$ and $M>1$, then $Q$ contains a compact invariant set which factors over a shift on $M$ symbols.References
- Keith Burns and Howard Weiss, A geometric criterion for positive topological entropy, Comm. Math. Phys. 172 (1995), no. 1, 95–118. MR 1346373, DOI 10.1007/BF02104512
- M. Carbinatto, J. Kwapisz, and K. Mischaikow, Horseshoes and the Conley index spectrum, Ergodic Theory Dynam. Systems 20 (2000), 365–377.
- M. C. Carbinatto and K. Mischaikow, Horseshoes and the Conley index spectrum. II. The theorem is sharp, Discrete Contin. Dynam. Systems 5 (1999), no. 3, 599–616. MR 1696332, DOI 10.3934/dcds.1999.5.599
- J. Kennedy, S. Koçak, and J. Yorke, A Chaos Lemma, to appear, Amer. Math. Monthly.
- J. Kennedy, M. Sanjuan, J. Yorke, and C. Grebogi, The topology of fluid flow past a sequence of cylinders, Topology and its Applications 94 (1999), 207–242.
- K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
- Konstantin Mischaikow and Marian Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math. 12 (1995), no. 2, 205–236. MR 1337206, DOI 10.1007/BF03167289
- Andrzej Szymczak, The Conley index and symbolic dynamics, Topology 35 (1996), no. 2, 287–299. MR 1380498, DOI 10.1016/0040-9383(95)00029-1
- Miguel A. F. Sanjuan, Judy Kennedy, Celso Grebogi, and James A. Yorke, Indecomposable continua in dynamical systems with noise: fluid flow past an array of cylinders, Chaos 7 (1997), no. 1, 125–138. MR 1439811, DOI 10.1063/1.166244
Additional Information
- Judy Kennedy
- Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
- Email: jkennedy@math.udel.edu
- James A. Yorke
- Affiliation: Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
- Email: yorke@ipst.umd.edu
- Received by editor(s): March 16, 1998
- Received by editor(s) in revised form: December 21, 1998
- Published electronically: February 15, 2001
- Additional Notes: This research was supported by the National Science Foundation, Division of Mathematical Sciences
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 2513-2530
- MSC (1991): Primary 58F12, 54F20; Secondary 54F50, 58F20
- DOI: https://doi.org/10.1090/S0002-9947-01-02586-7
- MathSciNet review: 1707195