Invariant Measures for Set-Valued Dynamical Systems

Miller, Walter; Akin, Ethan

doi:10.1090/S0002-9947-99-02424-1

Invariant Measures for Set-Valued Dynamical Systems
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by Walter Miller and Ethan Akin PDF

Trans. Amer. Math. Soc. 351 (1999), 1203-1225 Request permission

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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