Topological horseshoes (Q2706622)

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.