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Dynamics of induced systems

Published online by Cambridge University Press:  12 May 2016

ETHAN AKIN ,
JOSEPH AUSLANDER  and
ANIMA NAGAR
ETHAN AKIN
Affiliation:
Mathematics Department, The City College, 137 Street and Convent Avenue, New York, NY 10031, USA email [email protected]
JOSEPH AUSLANDER
Affiliation:
Mathematics Department, University of Maryland, College Park, MD 20742, USA email [email protected]
ANIMA NAGAR
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India email [email protected]
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Abstract

In this paper we study the dynamical properties of actions on the space of compact subsets of the phase space. More precisely, if $X$ is a metric space, let $2^{X}$ denote the space of non-empty compact subsets of $X$ provided with the Hausdorff topology. If $f$ is a continuous self-map on $X$, there is a naturally induced continuous self-map $f_{\ast }$ on $2^{X}$. Our main theme is the interrelation between the dynamics of $f$ and $f_{\ast }$. For such a study, it is useful to consider the space ${\mathcal{C}}(K,X)$ of continuous maps from a Cantor set $K$ to $X$ provided with the topology of uniform convergence, and $f_{\ast }$ induced on ${\mathcal{C}}(K,X)$ by composition of maps. We mainly study the properties of transitive points of the induced system $(2^{X},f_{\ast })$ both topologically and dynamically, and give some examples. We also look into some more properties of the system $(2^{X},f_{\ast })$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 7 , October 2017 , pp. 2034 - 2059
Copyright
© Cambridge University Press, 2016 

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