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. ©1999 American Mathematical Society.
Recommended Citation
W. Miller and E. Akin, "Invariant Measures for Set-valued Dynamical Systems," Transactions of the American Mathematical Society, vol. 351, no. 3, pp. 1203 - 1225, American Mathematical Society, Jan 1999.
The definitive version is available at https://doi.org/10.1090/s0002-9947-99-02424-1
Department(s)
Mathematics and Statistics
Keywords and Phrases
Basic set; Chain recurrence; Dynamics of a relation; Invariant measure; Sample path spaces; Set-valued dynamical system
International Standard Serial Number (ISSN)
0002-9947
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2026 American Mathematical Society, All rights reserved.
Publication Date
01 Jan 1999
