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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).
##### MSC:
 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)