On the existence and approximation of invariant densities for nonsingular transformations on \(\mathbb{R}^ d\)
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, \(C^ 2\) 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 transformation on \(\mathbb{R}^ d\) using the approximating sequence of Ulam.
nonsingular transformation on \(\mathbb{R}^ d\)
scientific article; zbMATH DE number 701333