Iterated means dichotomy for discrete dynamical systems (Q2301833)

The author discusses the dichotomy of iterated means for a compact discrete dynamical system acting on a finite-dimensional space, where \(k\)-order mean is a sequence generated by applying Cesaro averages \(k\)-times. The main result is the following statement. Let \(X\) be a convex compact subset of the finite dimensional space \(\mathbb{R}^m\) and \(\{x_j \}_{j=1}^{\infty}\subset X\) be any sequence. Then the following dichotomy holds, i.e., precisely one of the following two cases applies: (i) A \(k\)-order mean is convergent for any \(k\in \mathbb{N};\) (ii) A \(k\)-order mean is divergent for any \(k\in \mathbb{N}.\) As an application, the mean ergodicity of non-homogeneous Markov chains is investigated.