ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 68,   2   (1999)
pp.   205-211

ON TOPOLOGICAL SEQUENCE ENTROPY AND CHAOTIC MAPS ON INVERSE LIMIT SPACES
J. S. CANOVAS

Abstract.  The aim of this paper is to prove the following results: a continuous map $% f\:[0,1]\rightarrow [0,1]$ is chaotic iff the shift map $\sigma _f\:\lim\limits_\leftarrow ([0,1],f)\rightarrow \lim\limits_\leftarrow ([0,1],f)$ is chaotic. However, this result fails, in general, for arbitrary compact metric spaces. $\sigma _f\:\lim\limits_\leftarrow ([0,1],f)\rightarrow \lim\limits_\leftarrow ([0,1],f)$ is chaotic iff there exists an increasing sequence of positive integers $A$ such that the topological sequence entropy $h_A(\sigma _f)>0$. Finally, for any $A$ there exists a chaotic continuous map $f_A\:[0,1]\rightarrow [0,1]$ such that $% h_A(\sigma _f_A)=0.$

AMS subject classification.  58F03, 26A18
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