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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online