Absolutely continuous invariant measures for piecewise expandingC 2 transformations inR N

LetS be a bounded region inR ^N and let ℊ={S _i} _i ^=1/m be a partition ofS into a finite number of subsets having piecewiseC ² boundaries. We assume that whereC ² segments of the boundaries meet, the angle subtended by tangents to these segments at the point of contact is bounded away from 0. Letτ:S →S be piecewiseC ² on ℊ and expanding in the sense that there exists 0<σ< 1 such that for anyi=1, 2, ...,m, ‖Dτ _i ⁻¹ ‖<σ, whereDτ _i ⁻¹ is the derivative matrix ofτ _i ⁻¹ and ‖ ‖ is the euclidean matrix norm. The main result provides an upper bound onσ which guarantees the existence of an absolutely continuous invariant measure forτ. The research of the second author was supported by NSERC and FCAR grants.