This preview shows page 1. Sign up to view the full content.
Ergodic Theory
MTG6401
Homework5
Prof. JLF King
20Nov2006
Below,
(
T
:
X,
X
,μ
)
and
(
S
:
Y,
Y
,ν
)
are bimpts on
probability spaces. Say that
T
and
S
are
“
disjoint
in the sense of Furstenberg
”
if the space of joinings
J
(
T,S
) has but one point,
μ
×
ν
. We write
T
⊥
S
to
indicate that
T
and
S
are disjoint.
It is not di±cult to see that
T
⊥
S
implies that
T
and
S
are coprime. For if they had isomorphic
factors, then the relative independent joining over this
factor would be a nonproductmeasure joining of
T
with
S
.
Symmetric powers.
Let
T
×
n
mean the cartesian
n
th
power of
T
, that is,
T
×
n
...
×
T
.
Let
T
±
n
mean the
symmetric cartesian
n
th

power
of
T
. It is
T
×
n
/
≡
where two points
~x,~
y
∈
X
×
n
are equivalent,
~x
≡
~
y
, i² there is a permutation
π
of
[
1
..n
]
so that each
y
j
=
x
π
(
j
)
. Thus
T
±
n
is a fac
tor of
T
×
n
, and ﬁbers have
n
! many points.
General Notation.
This is the end of the preview. Sign up
to
access the rest of the document.
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

Click to edit the document details