Absolutely continuous invariant measures for piecewise expanding C-2 transformations in R(n) on domains with cusps on the boundaries

Let Omega be a bounded region in R(n) and let P = {Pi}(m)(i=1) be a partition of Omega into a finite number of subsets having piecewise C-2 boundaries. The boundaries may contain cusps. Let tau : Omega --> Omega be piecewise C-2 on P and expanding in the sense that there exists alpha > 1 such that for any i = 1, 2,..., m, parallel to D tau(i)(-1)parallel to < alpha(-1), where D tau(i)(-1) is the derivative matrix of tau(i)(-1) and parallel to .parallel to is the euclidean matrix norm. The main result provides a lower bound on alpha which guarantees the existence of an absolutely continuous invariant measure for tau.