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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Invariant Measures for
Set-Valued Dynamical Systems


Authors: Walter Miller and Ethan Akin
Journal: Trans. Amer. Math. Soc. 351 (1999), 1203-1225
MSC (1991): Primary 54H20, 58F10, 34C35
MathSciNet review: 1637090
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Abstract: A continuous map on a compact metric space, regarded as a dynamical system by iteration, admits invariant measures. For a closed relation on such a space, or, equivalently, an upper semicontinuous set-valued map, there are several concepts which extend this idea of invariance for a measure. We prove that four such are equivalent. In particular, such relation invariant measures arise as projections from shift invariant measures on the space of sample paths. There is a similarly close relationship between the ideas of chain recurrence for the set-valued system and for the shift on the sample path space.


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Additional Information

Walter Miller
Affiliation: Department of Mathematics, Howard University, Washington, D.C. 20059

Ethan Akin
Affiliation: Department of Mathematics, The City College, New York, New York 10031

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02424-1
PII: S 0002-9947(99)02424-1
Keywords: Set-valued dynamical system, dynamics of a relation, sample path spaces, invariant measure, basic set, chain recurrence
Received by editor(s): June 14, 1996
Article copyright: © Copyright 1999 American Mathematical Society