Jointly ergodic measure-preserving transformations Academic Article uri icon

abstract

  • The notion of ergodicity of a measure-preserving transformation is generalized to finite sets of transformations. The main result is that ifT 1,T 2, …,T s are invertible commuting measure-preserving transformations of a probability space (X, ℬ, μ) then \frac1N - Mån = MN - 1 T1n f1 .T2n f2 .....Tsn fs \xrightarrow[N - M ® µ ]I2 (X)(òX f1dm) (òX f2dm)...(òX fsdm) \frac{1}{{N - M}}\sum\limits_{n = M}^{N - 1} {T{}_1^n } f_1 .T_2^n f_2 .....T_s^n f_s \xrightarrow[{N - M \to \propto }]{{I^2 (X)}}(\int_X {f1d\mu )} (\int_X {f2d\mu )...(\int_X {fsd\mu )} } (1) for anyf 1,f 2, …,f s∈L x (X, ℬ, μ) iffT 1×T 2×…×T s and all the transformationsT iTj 1,i≠j, are ergodic. The multiple recurrence theorem for a weakly mixing transformation follows as a special case.

publication date

  • January 1, 1984