Jointly ergodic measure-preserving transformations |
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Authors: | Daniel Berend Vitaly Bergelson |
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Institution: | 1. Department of Mathematics, University of California, 90024, Los Angeles, CA, USA 2. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
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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, ?, μ) then 1 $$\frac{1}{{N - M}}\sum\limits_{n = M}^{N - 1} {T{}_1^n } f_1 .T_2^n f_2 .....T_s^n f_s \xrightarrow{N - M \to \propto }]{{I^2 (X)}}(\int_X {f1d\mu )} (\int_X {f2d\mu )...(\int_X {fsd\mu )} } $$ for anyf 1,f 2, …,f s∈L x (X, ?, 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. |
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