The problem of classification of triangular maps with zero topological entropy
Abstract
Relations between the following properties of triangular maps F: I2→I2 are studied in this paper: (P1) the period of any cycle is a power of two; (P2) every cycle is simple; (P3) the topological entropy of F restricted to the set of periodic points is 0; (P4) the topological entropy of F is 0; (P5) every ω-limit set contains a unique minimal set; (P6) F has no homoclinic trajectory; (P7) every ω-limit set either is a cycle or contains no cycle; (P8) no infinite ω-limit set contains a cycle. It is known that for continuous maps of the interval these properties are mutually equivalent. In the case of triangular maps of the square we prove that the properties (P1), (P2) and (P3) are mutually equivalent, and that (P7) is equivalent to (P8). Moreover, we show which of the remaining implications are true and which not. The problem is completely solved, with the following exceptions: we conjecture that (P7)⇒(P6) but we do not provide the argument; validity of (P7)⇒(P1) remains open. Our paper gives a partial solution of the problem stated in 1989 by A.N. Sharkovsky.
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Institute of Mathematics and Informatics, Silesian University in Opava, Czech Republic Czechia
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