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Abstract
Let $f$ be a continuous map $f:X\to X$ of a metric space $X$
into itself. Often the information about the map is presented in the
following form: for a finite collection of compact sets $A_1, \ldots,
A_n$ it is known which sets have the images containing other sets, and
which sets are disjoint. We study similar but weaker than usual
conditions on compact sets $A_1, \ldots, A_n$ assuming that the common
intersection of all sets $A_1,\ldots, A_n$ is empty (or making even
weaker but more technical assumptions). As we show, this implies that
the map is chaotic in the sense that it has positive topological
entropy, and moreover, there exists an invariant compact set on which
$f$ is semiconjugate to a full one-sided shift.
Mathematics Subject Classification: 30C35, 54F20.
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