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A triangular map with homoclinic orbits and no infinite \(\omega\)-limit set containing periodic points. (English) Zbl 1094.37019

Summary: Recently, G.-L. Forti, L. Paganoni and J. Smítal [Topology Appl. 153, 818–832 (2005; Zbl 1176.37010)] constructed an example of a triangular map of the unite square, \(F(x,y)=(f(x),g(x,y))\), possessing periodic orbits of all periods and such that no infinite \(\omega\)-limit set of \(F\) contains a periodic point. In this note, we show that the above quoted map \(F\) has a homoclinic orbit. As a consequence, we answer in the negative the problem presented by A.N. Sharkovsky in the eighties whether for a triangular map of the square, existence of a homoclinic orbit implies the existence of an infinite \(\omega\)-limit set containing a periodic point. It is well known that for a continuous map of the interval, the answer is positive.

MSC:

37E05 Dynamical systems involving maps of the interval
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37B40 Topological entropy
37B55 Topological dynamics of nonautonomous systems
26A18 Iteration of real functions in one variable
54H20 Topological dynamics (MSC2010)
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics

Citations:

Zbl 1176.37010
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References:

[1] Chudziak, J.; Snoha, Ľ.; Špitalský, V., From a Floyd-Auslander minimal system to an odd triangular map, J. Math. Anal. Appl., 296, 393-402 (2004) · Zbl 1046.37022
[2] Forti, G.-L.; Paganoni, L.; Smítal, J., Dynamics of homeomorphisms on minimal sets generated by triangular mappings, Bull. Austral. Math. Soc., 59, 1-20 (1999) · Zbl 0976.54043
[3] Forti, G.-L.; Paganoni, L.; Smítal, J., Triangular maps with all periods and no infinite \(ω\)-limit set containing periodic points, Topology Appl., 153, 5-6, 818-832 (2005) · Zbl 1176.37010
[4] Kočan, Z., The problem of classification of triangular maps with zero topological entropy, Ann. Math. Silesian., 13, 181-192 (1999) · Zbl 0988.37049
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[6] Sharkovsky, A. N.; Kolyada, S. F.; Sivak, A. G.; Fedorenko, V. V., Dynamics of One-Dimensional Mappings, Mathematics and its Applications (1997), Kluwer Academic: Kluwer Academic Dordrecht, (in Russian in 1989) · Zbl 0881.58020
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