Lyapunov functions and attractors

in arbitrary metric spaces

Author:
Mike Hurley

Journal:
Proc. Amer. Math. Soc. **126** (1998), 245-256

MSC (1991):
Primary 58F12

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 is an attractor for a continuous map of a metric space to itself, then there is a Lyapunov function for . 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:
http://dx.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