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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Topological horseshoes


Authors: Judy Kennedy and James A. Yorke
Journal: Trans. Amer. Math. Soc. 353 (2001), 2513-2530
MSC (1991): Primary 58F12, 54F20; Secondary 54F50, 58F20
Published electronically: February 15, 2001
MathSciNet review: 1707195
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02586-7
PII: S 0002-9947(01)02586-7
Keywords: Topological horseshoe, geometric horseshoe, chaos, shift dynamics, connection, preconnection, crossing number, expander, symbol set.
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
Article copyright: © Copyright 2001 American Mathematical Society



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