New results on topological dynamics of antitriangular maps (Q2756145)

The authors call a map \(F:[0,1]^{2}\to [0,1]^{2}\) antitriangular, if there are continuous maps \(f,g:[0,1]\to [0,1]\) such that \(F(x,y)=(g(y),f(x))\) for all \((x,y)\in [0,1]^{2}\). In this paper the relation between different sets of ``recurrent points'' (periodic points, uniformly recurrent points, recurrent points, centre, \(\omega\)-limits, non-wandering points, chain-recurrent points) is investigated. Obviously the dynamics of \(F\) is closely related to the dynamics of \(g\circ f\) and \(f\circ g\). Some relations between ``recurrent points'' of \(F\) and ``recurrent points'' of \(g\circ f\) and \(f\circ g\) are proved, for example \(\text{P}(F)= \text{P}(g\circ f)\times\text{P}(f\circ g)\) where \(\text{P}(F)\) is the set of periodic points of \(F\), or \(\text{UR}(F)\subseteq\text{UR}(g\circ f)\times\text{UR}(f\circ g)\) where \(\text{UR}(F)\) denotes the set of uniformly recurrent points of \(F\). In the latter case an example is given where \(\text{UR}(F)\neq\text{UR} (g\circ f)\times\text{UR}(f\circ g)\). For the non-wandering set \(\Omega (F)\) the authors give an example where \(\Omega (F)\subseteq\Omega (g\circ f)\times \Omega (f\circ g)\) does not hold.NEWLINENEWLINENEWLINEAn example of an antitriangular map is given where the sets of different types of ``recurrent points'' are different. Recently \textit{F. Balibrea} and \textit{A. Linero} [Periodic structure of \(\sigma\)-permutations maps on \(I^{n}\), Aequationes Math. 62, 265-279 (2001; Zbl 0991.37016)] proved that there exists an analogue of Sharkovskij's theorem for antitriangular maps. If \(F\) is ``simple'' in the sense (of the analogue) of Sharkovskij, then the authors prove that the sets of different types of ``recurrent points'' coincide. Moreover, it is shown that for any antitriangular map the centre equals the closure of the periodic points, and therefore it equals also the closure of the recurrent points.