Existence of many ergodic absolutely continuous invariant measures for piecewise-expandingC 2 chaotic transformations in ℝ2 on a fixed number of partitions

Abstract

Let Ω be a region in ℝn and letp = Pi )i1m, be a partition ofΩ into a finite number of closed subsets having piecewise C2 boundaries of finite(n - 1 )dimensional measure. Let τ:Ω→Ω be piecewise C2 onP where, τi = τ¦pi is aC2 diffeomorphism onto its image, and expanding in the sense that there exists α > 1 such that for anyi = 1, 2,...,m ‖Dτi-1 ‖ < α-1, where Dτi-1 is the derivative matrixτi- 1 and ¦‖·‖ is the Euclidean matrix norm. By means of an example, we will show that the simple bound of one-dimensional dynamics cannot be generalized to higher dimensions. In fact, we will construct a piecewise expanding C2 transformation on a fixed partition with a finite number of elements in ℝ2, but which has an arbitrarily large number of ergodic, absolutely continuous invariant measures