$napis="Abstract"; include "zacatek.inc"; ?> Preprint MA 52/2005
It is well-known that, for a continuous map $\varphi$ of the interval,
the condition {P1} $\varphi$ has zero topological entropy, is
equivalent, e.g., to any of the following: { P2} any $\omega
$-limit set contains a unique minimal set; { P3} the period of any
cycle of $\varphi$ is a power of two; { P4} any $\omega$-limit set
either is a cycle or contains no cycle; {P5} if $\omega
_\varphi(\xi)=\omega_{\varphi^2}(\xi)$, then $\omega_\varphi (\xi)$
is a fixed point; {P6} $\varphi $ has no homoclinic trajectory; {P7}
there is no countably infinite $\omega$-limit set; {P8}
trajectories of any two points are correlated; {P9} there is no
closed invariant subset $A$ such that $\varphi ^m|A$ is
topologically almost conjugate to the shift, for some $m\ge 1$. In
the paper we exhibit the relations between these properties in the
class $(x,y)\mapsto (f(x),g_x(y))$ of triangular maps of the square.
This contributes to the solution of a longstanding open problem of
Sharkovsky.
include "konec.inc";
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