Balibrea, F.; Smítal, J. A triangular map with homoclinic orbits and no infinite \(\omega\)-limit set containing periodic points. (English) Zbl 1094.37019 Topology Appl. 153, No. 12, 2092-2095 (2006). 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. Cited in 3 Documents 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 Keywords:triangular map; homoclinic orbit; periodic orbits; periodic point Citations:Zbl 1176.37010 PDFBibTeX XMLCite \textit{F. Balibrea} and \textit{J. Smítal}, Topology Appl. 153, No. 12, 2092--2095 (2006; Zbl 1094.37019) Full Text: DOI 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 [5] Kolyada, S. F., On dynamics of triangular maps of the square, Ergodic Theory Dynamical Systems, 12, 749-768 (1992) · Zbl 0784.58038 [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.