ACTA MATHEMATICA UNIVERSITATIS COMENIANAE
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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