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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Invariant Measures—Existence
verfasst von : Jiu Ding, Aihui Zhou
Erschienen in: Statistical Properties of Deterministic Systems
Verlag: Springer Berlin Heidelberg
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The existence problem of a stationary density to a Frobenius-Perron operator will be investigated. Three general existence theorems for Markov operators are presented with the help of the concepts of compactness, quasi-compactness, and constrictiveness. A powerful spectral decomposition theorem for constrictive Markov operators will be given. Then we prove the existence results for three concrete classes of transformations. They are the class of piecewise
C
2
and stretching interval mappings, the class of piecewise convex mappings with a weak repellor, and the class of multi-dimensional piecewise
C
2
and expanding transformations.