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Topological horseshoes. (English) Zbl 0972.37011
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.

37B10Symbolic dynamics
37C70Attractors and repellers, topological structure
37C25Fixed points, periodic points, fixed-point index theory
54F50Spaces of dimension $\le 1$; curves, dendrites
37D45Strange attractors, chaotic dynamics
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