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Zbl 0972.37011
Kennedy, Judy; Yorke, James A.
Topological horseshoes.
(English)
[J] Trans. Am. Math. Soc. 353, No.6, 2513-2530 (2001). ISSN 0002-9947; ISSN 1088-6850/e

The paper presents the notion of horseshoe dynamics in the non-hyperbolic case. Let $X$ be a separable metric space, $Q$ be a compact locally connected subset of $X$, and let $f:Q\to X$ be continuous. It is assumed that $Q$ contains two disjoint compact subsets $end_0$ and $end_1$ which intersect every component of $Q$. The crossing number of $Q$ is defined as the largest number $M$ such that every connection (i.e. a compact connected subset of $Q$ which intersects both $end_0$ and $end_1$) contains at least $M$ mutually disjoint preconnections, where a preconnection is defined as a compact connected subset of $Q$ such that its image under $f$ is a connection. The main theorem states that if the crossing number of $Q$ is $\geq 2$ then there exists a closed invariant subset $Q_I$ of $Q$ for which $f|_{Q_I}$ is semiconjugated to the one-sided shift on $M$-symbols. Some examples and other related results are presented.
[R.Srzednicki (Kraków)]
MSC 2000:
*37B10 Symbolic dynamics
37C70 Attractors and repellers, topological structure
37C25 Fixed points, periodic points, fixed-point index theory
54F50 Spaces of dimension $\le 1$
37D45 Strange attractors, chaotic dynamics

Keywords: topological horseshoe; shift dynamics; chaos; crossing number

Cited in: Zbl 1189.37016 Zbl 1213.37125

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