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 writeup.)
Notation. On a normed vectorspace, the
operatornorm of a linear operator U: H→H,
written U op , is the supremum of Uv over all
unitvectors v ∈ H. If U op 1 then we call U
a weak contraction. Finally, use
FixU := {v ∈ H  Uv = v}
for the set of ﬁxedpoints of U.
On an innerproduct 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 circlegroup, 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 measuretheoretically isomorphic to Id × R, by producing an explicit (very simple)
bimp 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
measurepreserving. 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 nontrivial R×Rinvariant set. ...
View
Full Document
 Fall '09
 Staff
 Vector Space, Hilbert space, Prof. JLF King, weak contraction, Ergodic HW Zwei

Click to edit the document details