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Compactness of Invariant Densities for Families of Expanding, Piecewise Monotonic Transformations

Published online by Cambridge University Press:  20 November 2018

P. Góra  and
A. Boyarsky
P. Góra
Affiliation:
Warsaw University, Warsaw, Poland
A. Boyarsky
Affiliation:
Concordia University, Montreal, Quebec
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Let I = [0,1] and let be the space of all integrable functions on I, where m denotes Lebesque measure on I. Let ∥ ∥1 be the ℒ-1-norm and let be a measurable, nonsingular transformation on I. Let denote the space of densities. The probability measure μ is invariant under τ if for all measurable sets A, The measure μ is absolutely continuous if there exists an such that for any measurable set A We refer to ƒ* as the invariant density of τ (with respect to m). It is well-known that ƒ * is a fixed point of the Frobenius-Perron operator defined by

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 5 , 01 October 1989 , pp. 855 - 869
Copyright
Copyright © Canadian Mathematical Society 1989

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