Geodesic
Compactness of Invariant Densities for Families of Expanding, Piecewise Monotonic Transformations
Canadian journal of mathematics, Tome 41 (1989) no. 5, pp. 855-869

Voir la notice de l'article provenant de la source Cambridge University Press

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 
Góra, P.; Boyarsky, A. Compactness of Invariant Densities for Families of Expanding, Piecewise Monotonic Transformations. Canadian journal of mathematics, Tome 41 (1989) no. 5, pp. 855-869. doi: 10.4153/CJM-1989-039-8
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