# American Institute of Mathematical Sciences

March  2004, 11(2&3): 547-556. doi: 10.3934/dcds.2004.11.547

## The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval

 1 Department of General Education for the Hearing Impaired, Tsukuba College of Technology, Ibaraki 305-0005, Japan 2 Institute of Mathematics, University of Tsukuba, Ibraki 305-8571, Japan

Received  October 2002 Revised  February 2004 Published  June 2004

Let $f$ be a continuous map from the unit interval to itself. In this paper, it is shown that $f$ has positive topological entropy if and only if $f$ is pointwise $P$-expansive for some periodic orbit $P$ of $f$. And it is also proved that if $f$ has a periodic orbit with odd period, then there exists a chaotic map from a dendrite to itself in the sense of Devaney which is semiconjugate to $f$ and has positive topological entropy.
Citation: Tatsuya Arai, Naotsugu Chinen. The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 547-556. doi: 10.3934/dcds.2004.11.547
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