hw5.ergd - Ergodic Theory MTG 6401 Homework-5 Prof. JLF...

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Ergodic Theory MTG6401 Homework-5 Prof. JLF King 20Nov2006 Below, ( T : X, X ) and ( S : Y, Y ) are bi-mpts on probability spaces. Say that T and S are disjoint in the sense of Furstenberg if the space of joinings J ( T,S ) has but one point, μ × ν . We write T S to indicate that T and S are disjoint. It is not di±cult to see that T S implies that T and S are co-prime. For if they had isomorphic factors, then the relative independent joining over this factor would be a non-product-measure joining of T with S . Symmetric powers. Let T × n mean the cartesian n th -power of T , that is, T × n ... × T . Let T ± n mean the symmetric cartesian n th - power of T . It is T × n / where two points ~x,~ y X × n are equivalent, ~x ~ y , i² there is a permutation π of [ 1 ..n ] so that each y j = x π ( j ) . Thus T ± n is a fac- tor of T × n , and fibers have n ! many points. General Notation.
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This note was uploaded on 01/26/2012 for the course MTG 6401 taught by Professor Staff during the Fall '09 term at University of Florida.

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