hw1-ERG - Ergodic Theory MTG6401 Ergodic HW Uno Prof. JLF...

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Unformatted text preview: Ergodic Theory MTG6401 Ergodic HW Uno Prof. JLF King 29Jan2010 ( Due Wedn., 16Sept, at the beginning of class. Please staple this sheet as the first page of your write-up. ) Notation. Let K be the unit circle, [ , 1 ) wrapped into a circle, with m () its arclength mea- sure. For R , let R be rotation by on K . For explanation of your solutions to each of these three problems, pictures are appropriate. H1: Here ( T : X, X , ) is a mpt, and A is an invariant set, i.e, T 1 ( A ) a.e = A . a Suppose T is bi-mpt. Construct a set E a.e = A so that E is exactly-invariant , i.e, T 1 ( E ) = E . b No longer assume that T is invertible. With B := T k =0 T- k ( A ), prove that B a.e = A and T 1 ( B ) B . Now use B to construct a set E a.e = B which is exactly-invariant. H2: Let R = R be an irrational rotation on K . Partition the circle into two half-open intervals, say A := 0 , 1 3 ) and B := 1 3 , 1 ) ....
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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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