Invariant Measures for Set-Valued Dynamical Systems |
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Authors: | Walter Miller Ethan Akin |
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Institution: | Department of Mathematics, Howard University, Washington, D.C. 20059 Ethan Akin ; Department of Mathematics, The City College, New York, New York 10031 |
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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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Keywords: | Set-valued dynamical system dynamics of a relation sample path spaces invariant measure basic set chain recurrence |
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