**
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 w_{j}(x)
= w_{j2}(x),
then w_{j} (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 j^{m}|*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*), *g*_{x}(*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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