On the number of invariant measures for higher-dimensional chaotic transformations (Q1182255)

Let \(S\) be a bounded region in \(\mathbb{R}^ N\) and let \({\mathcal P}=\{S_ i\}^ m_{i=1}\) be a partition of \(S\) into a finite number of closed subsets having piecewise \(C^ 2\) boundaries of finite \((N-1)\)- dimensional measure. Let \(\tau: S\to S\) be piecewise \(C^ 2\) on \(\mathcal P\) and expanding in the sense that there exists \(0<\sigma < 1\) such that for any \(i=1,2,\dots,m\), \(\| DT_ i^{-1}\|<\sigma\), where \(DT_ i^{-1}\) is the derivative matrix of \(T_ i^{-1}\) and \(\| .\|\) is the Euclidean matrix norm. The authors prove that for some classes of such mappings, for example, Jablonski transformations or convexity- preserving transformations, the number of crossing points constitutes a bound for the number of ergodic, absolutely continuous \(\tau\)-invariant measures. Then the authors give examples showing that in general the simple bound of one-dimensional dynamics cannot be generalized to higher dimensions. In fact, the authors show that it is possible to construct piecewise expanding \(C^ 2\) transformations on a fixed partition with a finite number of elements but which have an arbitrarily large number of ergodic, absolutely continuous invariant measures.