hw2-ERG - Ergodic Theory MTG 6401 Ergodic HW Zwei Prof. JLF...

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Unformatted text preview: Ergodic Theory MTG 6401 Ergodic HW Zwei Prof. JLF King 3Oct2009 (Due Monday, 21Sep2009, hopefully. Please staple this sheet as the first page of your write-up.) Notation. On a normed vectorspace, the operator-norm of a linear operator U: H→H, written U op , is the supremum of Uv over all unit-vectors v ∈ H. If U op 1 then we call U a weak contraction. Finally, use FixU := {v ∈ H | Uv = v} for the set of fixed-points of U. On an inner-product space H, the “adjoint of U ” is the linear operator U that satisfies ∀v,w ∈ H : U v, w = v, Uw . H4: Henceforth U: H→H is a weak contraction on a Hilbert space. (Below, are two facts we used in our proof of the full L2 Ergodic Thm.) a Please prove that U is a weak contraction. (You are free to establish a stronger statement.) b Prove that FixU = FixU . H5: Fix a rotation number α ∈ R. Let X and Y denote copies of [0, 1), viewed (and topologized) as the circle-group, with ⊕ and denoting addition and subtraction mod 1. Use m() for arclength measure, and let R=Rα be the rotation x → x ⊕ α. Prove that R × R is measure-theoretically isomorphic to Id × R, by producing an explicit (very simple) bi-mp map f : X×Y →X×Y such that this diagram R ×R X × Y −−→ X × Y −− f f Id×R X × Y −−→ X × Y −− commutes. (Here, Id = IdX is the identity map on X .) The f that I’m imagining is a homeomorphism, and is “algebraic”. Make sure to prove that your f is measure-preserving. Please draw a picture showing how your f works. (By the way, how does your f vary as a function of α?) Finally, use your isomorphism to show that R × R is never ergodic (no matter what α is). Produce an explicit non-trivial R×R-invariant set. ...
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