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

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Ergodic Theory MTG 6401 Homework-5 Prof. JLF King 20Nov2006 Below, ( T : X, X , μ ) and ( S : Y, Y , ν ) 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 difficult 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 , iff 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. If f : X R and g : Y R , let f × g denote the fnc ( x, y ) 7→ f ( x ) g ( y ).
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