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February, 1985 Conjecture: In General a Mixing Transformation is Not Two-Fold Mixing
Steven Alpern
Ann. Probab. 13(1): 310-313 (February, 1985). DOI: 10.1214/aop/1176993084


A new topology is introduced on the group of $\mu$-preserving automorphisms of a Lebesgue space $(X, \Sigma, \mu)$ so that $f_k \rightarrow f$ if $\mu(f^n_kA \cap B) \rightarrow \mu(f^nA \cap B)$ uniformly in $n$ for all $A, B$ in $\Sigma$. The subspace of mixing automorphisms is a Baire space in the relative topology. A conjecture (about the extent to which a mixing stationary process is determined by its two-dimension distributions) is stated, which is true implies that the two-fold mixing automorphisms are of first category in the mixing ones. So if the conjecture is true then by Baire's Theorem there is a mixing but not two-fold mixing automorphism.


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Steven Alpern. "Conjecture: In General a Mixing Transformation is Not Two-Fold Mixing." Ann. Probab. 13 (1) 310 - 313, February, 1985.


Published: February, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0574.28012
MathSciNet: MR770646
Digital Object Identifier: 10.1214/aop/1176993084

Primary: 28D05

Keywords: Measure preserving transformation , Mixing , two-fold mixing

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 1 • February, 1985
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