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 Uinvariant vectors. Let
−
1
AN := N N=01 Uk . Then
k 95pts A3: A1: Recall the Mean Ergodic Thm. 115pts A2: ClassA 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 weakcontraction U, that a vector
w is U invariant IFF w is Uinvariant. (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 Rforwardorbit 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 base4. Let ∆(z ) ∈ [1 .. 3] denote
the highorder ﬁt (f ourary digit) of the basefour
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 ClassA Honor Code: “I have neither requested nor received
help on this exam other than from my teammates and my
professor (or his colleague).”
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This note was uploaded on 01/26/2012 for the course MTG 6401 taught by Professor Staff during the Fall '09 term at University of Florida.
 Fall '09
 Staff

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