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