a-cl-ERG - DynSys MTG 6401 Prof JLF King 4Dec2009 A1 1...

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Unformatted text preview: DynSys MTG 6401 Prof. JLF King 4Dec2009 A1: 1: L2 -MET. Suppose U: H→H is a weakcontraction, U op 1, on a Hilbert space. Let P be orthogonal projection (an operator) onto W, the (closed) subspace of U-invariant vectors. Let − 1 AN := N N=01 Uk . Then k 95pts A3: A1: Recall the Mean Ergodic Thm. 115pts A2: Class-A 25pts Total: 235pts Please PRINT your name and student ordinal; Ta: Ord: .............................................. SoT − AN −→ P , as N ∞. I.e, for each v ∈ H, necessarily AN (v) → P(v). ♦ a Prove L2 -MET when U is a unitary operator, U = U 1 . (In this and the next part, if you want to assume that U is the Koopman operator of a mpt, that is fine.) b Prove, for a weak-contraction U, that a vector w is U -invariant IFF w is U-invariant. (You may use for free that U op = U op .) Describe the modification to get the full L2 MET, even if U is not unitary. A2: Suppose R = Rα is an irrational rotation on the “circle” X := [0, 1). Prove that the R-forward-orbit of an arbitrary z ∈ X , is dense. (Please use O+ (z ) for the orbit.) A3: Short answer; show no work. Imagine we express points x ∈ R+ in base-4. Let ∆(z ) ∈ [1 .. 3] denote the high-order fit (f our-ary digit) of the base-four numeral of z . Let S :R+ be the doubling map, z → 2z . Thus 1 N N −1 ∆(S k z ) N →∞ −→ L, k=0 where L is the number ....................... . (Make sure to write the base of any logarithms you use.) End of Class-A Honor Code: “I have neither requested nor received help on this exam other than from my team-mates and my professor (or his colleague).” Signatures, dates: .............................................. ...
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