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 thefollowing form: for a finite collection of compact sets $A_1, \ldots,A_n$ it is known which sets have the images containing other sets, andwhich sets are disjoint. We study similar but weaker than usualconditions on compact sets $A_1, \ldots, A_n$ assuming that the commonintersection of all sets $A_1,\ldots, A_n$ is empty (or making evenweaker but more technical assumptions). As we show, this implies thatthe map is chaotic in the sense that it has positive topologicalentropy, and moreover, there exists an invariant compact set on which$f$ is semiconjugate to a full one-sided shift.
Highlights
The modern theory of topological dynamical systems studies maps f : X → X of topological spaces
In our case the positive entropy is assumed on an invariant for some power of f horseshoe-like set. This is related to a deep question whether for certain classes of dynamical systems the topological entropy of the map can be approximated on horseshoes admitted by its powers
Just like the existence of a weak horseshoe implies that there is a set on which a map is semiconjugate to a full one-sided shift, there are conditions which on the surface of it look weaker than that of existence of a weak horseshoe while imply such, perhaps for some power of f
Summary
The modern theory of topological dynamical systems studies (continuous) maps f : X → X of topological spaces (in a more general sense it studies flows, yet in this paper we concentrate upon maps only). In our case the positive entropy is assumed on an invariant for some power of f horseshoe-like set This is related to a deep question whether for certain classes of dynamical systems the topological entropy of the map can be approximated on horseshoes admitted by its powers. Even a relevant but weaker question below does not have a general answer: for what classes of non-invertible dynamical systems is it true that the existence of horseshoes in the above sense is equivalent to the positive entropy of the map? It may well be so that combining the tools developed in [KY] and our tools one can get new results, and we thank the referee for drawing our attention to the paper [KY] as well as for other useful remarks
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