Set-Valued Analysis

, Volume 3, Issue 2, pp 181–194

Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems

  • Walter M. Miller

DOI: 10.1007/BF01038599

Cite this article as:
Miller, W.M. Set-Valued Anal (1995) 3: 181. doi:10.1007/BF01038599


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.

Mathematics Subject Classifications (1991)

Primary: 58F11Secondary: 54C60, 60J05

Key words

Frobenius-Perron operatorsinvariant measuresergodic theoryset-valued dynamical systemsrelationsrandom perturbations

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Walter M. Miller
    • 1
  1. 1.Department of MathematicsHoward UniversityWashington, DCU.S.A.