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Hierarchies of chaotic maps on continua

Published online by Cambridge University Press:  13 August 2013

LOGAN HOEHN  and
CHRISTOPHER MOURON
LOGAN HOEHN
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, 1300 University Boulevard, Birmingham, AL 35294, USA Department of Mathematics, Nipissing University, 100 College Drive, Box 5002, North Bay, ON, Canada P1B 8L7 email [email protected]
CHRISTOPHER MOURON
Affiliation:
Department of Mathematics and Computer Science, Rhodes College, Memphis, TN 38112, USA email [email protected]
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Abstract

Let $f: X\longrightarrow X$ be a map of a continuum. In this paper we examine the following dynamical conditions on $f$: (1) $f$ is continuum-wise fully expansive; (2) $f$ is weakly continuum-wise fully expansive; (3) $f$ is mixing; (4) $f$ is weakly mixing. We first show that (1) implies (2), (2) implies (3) and (3) implies (4). Then we investigate what topological conditions will force the reverse implications to hold and give examples of when the reverse conditions do not hold. In particular, a map of the universal dendrite is given that is weakly mixing but not mixing.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 6 , December 2014 , pp. 1897 - 1913
Copyright
© Cambridge University Press, 2013 

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