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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

Tatsuya Arai 1, and  Naotsugu Chinen 2,

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.
Keywords: Sharkovsky ordering, topological entropy, Chaotic map in the sense of Devaney, pointwise P-expansive..
Mathematics Subject Classification: 54H20, 37B40, 37E05, 54F5.
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 and Continuous Dynamical Systems, 2004, 11 (2&3) : 547-556. doi: 10.3934/dcds.2004.11.547
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