Ergodic Theory MTG6401 Ergodic HW Prof. JLF King 20Nov2006 H1: Suppose ( T : X, X ,μ ) is a bi-mpt. and A is a ( measurable ) set such that T 1 ( A ) a.e = A . Prove that there exists a set B a.e = A so that B is pointwise invariant, i.e, T 1 ( B ) = B . H2: Let R = R α be an irrational rotation of the circle X := [0 , 1 ) . Partition the circle into two half-open intervals, say A := [0 , 1 3 ) and B := [ 1 3 , 1 ) . Given a point z ∈ X , let the forward name of z be the sequence ( s0 ,s 1 ,s 2 ,... ) , where s n is the symbol “
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