On the Existence and Approximation of Invariant Densities for Nonsingular Transformations on Rd

AbstractLet T be a piecewise monotone, expanding, and C2 mapping of the unit interval to itself which admits an absolutely continuous invariant measure ν = ƒ dm. S. Ulam has described a sequence of finite dimensional operators Pn approximating the Frobenius-Perron operator associated to T, and conjectured that the sequence of non-negative fixed points ƒn obtained for the Pn converge strongly to ƒ. 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, C2 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 Rd using the approximating sequence of Ulam