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Non-chaotic antitriangular maps. (English) Zbl 1092.37008
Summary: By an antitriangular map we understand a continuous map from \(I^2=[0,1]^2\) into itself with the form \(F(x,y)=(g(y),f(x))\). If \(f,g:I\to I\) are continuous piecewise monotone maps such that \(g\circ f\) and \(f \circ g\) have type \(2^\infty\), then we prove that \(F\) is not chaotic in the sense of Li-Yorke if and only if UR\((F)=\text{UR}(g\circ f)\times\text{UR}(f\circ g)\), where UR\(( \cdot)\) denotes the set of uniformly recurrent points. Moreover, we show that in the general case of nonchaotic antitriangular maps \(F\) (with coordinates not necessarily piecewise monotonic) we have \(\omega(F)=C(F)=\omega(g\circ f)\times \omega(f\circ g)\), and hence \(\omega(F)\) is a closed set \((\omega (\cdot)\) and \(C(\cdot)\) denote the sets of limit points and the centre of a continuous map, respectively).
37B20 Notions of recurrence and recurrent behavior in dynamical systems
37E99 Low-dimensional dynamical systems
54H20 Topological dynamics (MSC2010)
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)