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 | 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.

**[1]**Ethan Akin,*The general topology of dynamical systems*, Graduate Studies in Mathematics, vol. 1, American Mathematical Society, Providence, RI, 1993. MR**1219737****[2]**Jean-Pierre Aubin and Hélène Frankowska,*Set-valued analysis*, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR**1048347****[3]**Jean-Pierre Aubin, Hélène Frankowska, and Andrzej Lasota,*Poincaré’s recurrence theorem for set-valued dynamical systems*, Ann. Polon. Math.**54**(1991), no. 1, 85–91. MR**1132077****[4]**R. A. Brualdi,*Convex sets of non-negative matrices*, Canad. J. Math.**20**(1968), 144–157. MR**0219556****[5]**David Gale,*The theory of linear economic models*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960. MR**0115801****[6]**Ĭ. Ī. Gīhman and A. V. Skorohod,*The theory of stochastic processes. I*, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by S. Kotz; Die Grundlehren der mathematischen Wissenschaften, Band 210. MR**0346882****[7]**John G. Hocking and Gail S. Young,*Topology*, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR**0125557****[8]**John E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), no. 5, 713–747. MR**625600**, 10.1512/iumj.1981.30.30055**[9]**Richard McGehee,*Attractors for closed relations on compact Hausdorff spaces*, Indiana Univ. Math. J.**41**(1992), no. 4, 1165–1209. MR**1206344**, 10.1512/iumj.1992.41.41058**[10]**Walter M. Miller,*Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems*, Set-Valued Anal.**3**(1995), no. 2, 181–194. MR**1343483**, 10.1007/BF01038599

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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

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