Smoothness of invariant density for expanding transformations in higher dimensions

Abstract

Let Ω be a bounded region in ℝn and let 𝒫={Pi}i=1m be a partition of Ω into a finite number of closed subsets having piecewise C2 boundaries of finite (n−1)-dimensional measure. Let τ:Ω→Ω be an expanding transformation on 𝒫 where, τi:τ|Pi, and τi∈CM, m≥2. We show that the τ-invariant density h∈CM−2.