Vol. LXXV, 2 (2006)
p. 241 - 252

A classification of triangular maps of the square
V. Kornecka

Abstract.  It is well-known that, for a continuous map j of the interval, the condition P1 j has zero topological entropy, is equivalent, e.g., to any of the following: P2 any w-limit set contains a unique minimal set; P3 the period of any cycle of j is a power of two; P4 any w-limit set either is a cycle or contains no cycle; P5 if wj(x) = wj2(x), then wj (x) is a fixed point; P6 j has no homoclinic trajectory; P7 there is no countably infinite w-limit set; P8 trajectories of any two points are correlated; P9 there is no closed invariant subset A such that jm|A is topologically almost conjugate to the shift, for some m ³ 1. In the paper we exhibit the relations between these properties in the class (x,y) ®(f(x), gx(y)) of triangular maps of the square. This contributes to the solution of a longstanding open problem of Sharkovsky.

Keywords. Triangular map, topological entropy, w-limit set.  

AMS Subject classification.  Primary: 37B20, 37B40, 54H20.

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