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

DOI:
https://doi.org/10.1090/S0002-9947-99-02424-1

MathSciNet review:
1637090

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9947-99-02424-1

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