ACTA MATHEMATICA
UNIVERSITATIS COMENIANAE




Pre-image entropy for maps on noncompact topological spaces

Lei Liu

Received: August 24, 2012;   Accepted: April 16, 2013



Abstract.   We propose a new definition of pre-image entropy for continuous maps on noncompact topological spaces, investigate fundamental properties of the new pre-image entropy, and compare the new pre-image entropy with the existing ones. The defined pre-image entropy generates that of Cheng and Newhouse. Yet, it holds various basic properties of Cheng and Newhouse's pre-image entropy, for example, the pre-image entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have the same pre-image entropy, the pre-image entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new pre-image entropy coincides with Cheng and Newhouse's pre-image entropy for compact systems.

Keywords:  Pre-image entropy; Locally compact space; Alexandroff compactification; Hyperspace dynamical system.  

AMS Subject classification: Primary:  54H20, 28D20 


Version to read:   PDF






Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

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
© 2013, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE