ON THE EXISTENCE AND APPROXIMATION OF INVARIANT DENSITIES FOR NONSINGULAR TRANSFORMATIONS ON R(D)

[摘要] Let T be a piecewise monotone, expanding, and C-2 mapping of the unit interval to itself which admits an absolutely continuous invariant measure nu = f dm. S. Ulam has described a sequence of finite dimensional operators P-n approximating the Frobenius-Perron operator associated to T, and conjectured that the sequence of non-negative fixed points f(n) obtained for the P-n converge strongly to f. This was shown to be the case by T. Y. Li. A. Boyarsky and S. Y. Lou gave a partial generalization of this result to the case of expanding, C-2 Jablonski transformations on the multidimensional unit cube, obtaining weak approximation of the invariant density. In this article we replace weak with strong convergence in the multidimensional result using a compactness criterion due to Kolmogorov. We also discuss both existence and approximation of the invariant density in the case of general nonsingular transformations on R(d) using the approximating sequence of Ulam. (C) 1994 Academic Press, Inc.

[发布日期] 1994-11-01