Journal of Applied Mathematics and Stochastic Analysis 
Volume 13 (2000), Issue 1, Pages 33-40
doi:10.1155/S1048953300000046

Smoothness of invariant density for expanding transformations in higher dimensions

Kourosh Adl-Zarabi1,2 and Harald Proppe2

 1Dawson College, Department of Mathematics, 3040 Sherbrooke St. West, Montreal H3Z 1A4, Canada
2Concordia University, Department of Mathematics and Statistics, 7141 Sherbrooke St. West, Montreal H4B 1R6, Canada
Received 1 August 1998; Revised 1 September 1999

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.