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rank1-zeroentr

# rank1-zeroentr - Step 2 Fix a sequence n DTASARenumbering...

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Rank- 1 has zero entropy Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 20 November, 2006 (at 18:39 ) Using stacks Let ---→ STKs := ( Ξ n ) n =1 be the stacks used to cut&stack a rank-1 T , on a non-atomic Lebesgue space ( X, X , μ ) . Let L n denote the height of Ξ n . Let S n X denote the spacers adjoined to make the n -stack. Thus Ξ n t S n +1 = Ξ n +1 . Let A n := F j = n +1 S j be the spacers adjoined a fter stage- n . Certainly μ ( A n ) & 0. I first construct a particular 2-set generating parti- tion P = ( B, G ) . Step 1. For ( Ξ n ) n =1 , I will DTASARenumber ( Drop To A Subsequence And Renumber ) several times, so as to gain a new property for ---→ STKs. Later subse- quencings will preserve the properties obtained ear- lier. I can initially DTASARenumber so that L n > n + 3, for all n . First DTASARenumber so that there are at least 2 n copies of Ξ n in Ξ n +1 . Now simply declare that the bottom-most copy of Ξ n in Ξ n +1 is, in fact, spacer which is part of S n +1 . Since

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rank1-zeroentr - Step 2 Fix a sequence n DTASARenumbering...

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