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How little is little enough?

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  • 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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