ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

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
Keywords.

Download: Adobe PDF

Compressed Postscript

Acta Mathematica Universitatis Comenianae
Institute of Applied Mathematics
Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic
Telephone: + 421-2-60295111 Fax: + 421-2-65425882
e-Mail: amuc@fmph.uniba.sk Internet: www.iam.fmph.uniba.sk/amuc