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Complicated dynamics for low-dimensional strongly monotone maps. (English) Zbl 0893.54034

Let \(X\) be a strongly ordered Banach space, \(P:X\to X\) be a completely continuous, \(C^1\), point-dissipative map whose derivative is strongly positive at every point of \(X\). Then there is a positive integer \(q\) and an open dense set of \(U\subset X\) such that the omega limit set of every point of \(U\) is a periodic orbit with at most \(q\) (Tereščák, 1996). In the paper some examples which show that this result is best possible are given. It is shown that most one-dimensional maps can be imbedded in a planar monotone map. This implies that planar monotone maps can have complicated dynamics at least on a proper invariant subspace of the plane. These ideas are extended to higher dimensions. There is a map on \(\mathbb{R}^4\) satisfying the hypotheses of Tereščák’s theorem which has a hyperbolic invariant horseshoe on \(\Gamma= \{(u,v): u=-v\}\).

MSC:

54H20 Topological dynamics (MSC2010)
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