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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 $en{d}_{0}$ and $en{d}_{1}$ 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 $en{d}_{0}$ and $en{d}_{1}$) 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 $\ge 2$ 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.

##### MSC:
 37B10 Symbolic dynamics 37C70 Attractors and repellers, topological structure 37C25 Fixed points, periodic points, fixed-point index theory 54F50 Spaces of dimension $\le 1$; curves, dendrites 37D45 Strange attractors, chaotic dynamics
##### Keywords:
topological horseshoe; shift dynamics; chaos; crossing number