On the number of invariant measures for higher-dimensional chaotic transformations

Abstract

LetS be a bounded region inR N and letP={S l } i=1 m be a partition ofS into a finite number of closed subsets having piecewiseC2 boundaries of finite (N−1)-dimensional measure. Let τ:S→S be piecewiseC2 onP and expanding in the sense that there exists 0<σ<1 such that for anyi=1,2,...,m, ∥DT i −1∥<σ, whereDT i −1 is the derivative matrix ofT i −1 and ∥·∥ is the Euclidean matrix norm. We prove that for some classes of such mappings, for example, Jabtonski transformations or convexity-preserving transformations, the number of crossing points constitutes a bound for the number of ergodic absolutely continuous τ-invariant measures. We give examples showing that in general the simple bound of one-dimensional dynamics cannot be generalized to higher dimensions. In fact, we show that it is possible to construct piecewise expandingC2 transformations on a fixed partition with a finite number of elements but which have an arbitrarily large number of ergodic, absolutely continuous invariant measures.