A Compactness Theorem for Approximating the Invariant Densities of Higher Dimensional Transformations

Let τ be a Jablonski transformation from the n-dimensional unit cube U into itself which has a unique absolutely continuous invariant measure with density function ƒ. Let T denote a family of transformations which approximate τ on finer and finer partitions. The main result of this paper is a compactness theorem on the densities associated with T which allows us to prove that the invariant densities associated with the transformations in T converge weakly to ƒ.