Jointly ergodic measure-preserving transformations
The notion of ergodicity of a measure preserving transformation is generalized to finite sets of transformations. The main result is that, if \(T_ 1,T_ 2,...,T_ s\) are invertible commuting measure preserving transformations, of a probability space (X,\({\mathcal B},\mu)\), then \[ \frac{1}{N-M}\sum^{N-1}_{n=M}T^ n_ 1f_ 1\cdot T^ n_ 2f_ 2\cdot...\cdot T^ n_ sf_ s\to^{L^ 2(X)}_{N-M\to \infty}\int_{X}f_ 1d\mu \cdot \int_{X}f_ 2d\mu \cdot...\cdot \int_{X}f_ sd\mu \] for every \(f_ 1,f_ 2,...,f_ s\in L^{\infty}(X,{\mathcal B},\mu)\) iff \(T_ 1\times T_ 2\times...\times T_ s\) and all the transformations \(T_ iT_ j^{-1}\), \(i\neq j\), are ergodic. The multiple recurrence theorem for a weakly mixing transformation follows as a special case. For more results, especially in the case where the system in question consists of a compact abelian group and endomorphisms thereof, see: \textit{D. Berend}, Joint Ergodicity and Mixing, J. d'Analyse Math. (to appear).
ergodicity of a measure preserving transformation
multiple recurrence theorem for a weakly mixing transformation
scientific article; zbMATH DE number 3910702
scientific article