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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Lyapunov functions and attractors
in arbitrary metric spaces


Author: Mike Hurley
Journal: Proc. Amer. Math. Soc. 126 (1998), 245-256
MSC (1991): Primary 58F12
DOI: https://doi.org/10.1090/S0002-9939-98-04500-6
MathSciNet review: 1458880
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Abstract: We prove two theorems concerning Lyapunov functions on metric spaces. The new element in these theorems is the lack of a hypothesis of compactness or local compactness. The first theorem applies to a discrete dynamical system on any metric space; the result is that if $A$ is an attractor for a continuous map $g$ of a metric space $X$ to itself, then there is a Lyapunov function for $A$. The second theorem applies only to separable metric spaces; the theorem is that there is a complete Lyapunov function for any continuously-generated discrete dynamical system on a separable metric space. (A complete Lyapunov function is a real-valued function that is constant on orbits in the chain recurrent set, is strictly decreasing along all other orbits, and separates different components of the chain recurrent set.)


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Additional Information

Mike Hurley
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106-7058
Email: mgh3@po.cwru.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04500-6
Keywords: Attractor, Lyapunov function, chain recurrence
Received by editor(s): May 12, 1994
Communicated by: Mary Rees
Article copyright: © Copyright 1998 American Mathematical Society

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