Abstract
A set-valued dynamical systemF on a Borel spaceX induces a set-valued operatorF onM(X) — the set of probability measures onX. We define arepresentation ofF, each of which induces an explicitly defined selection ofF; and use this to extend the notions of invariant measure and Frobenius-Perron operators to set-valued maps. We also extend a method ofS. Ulam to Markov finite approximations of invariant measures to the set-valued case and show how this leads to the approximation ofT-invariant measures for transformations τ, whereT corresponds to the closure of the graph of τ.
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Miller, W.M. Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems. Set-Valued Anal 3, 181–194 (1995). https://doi.org/10.1007/BF01038599
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DOI: https://doi.org/10.1007/BF01038599