Rank
1
has zero entropy
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
[email protected]
Webpage
http://www.math.ufl.edu/
∼
squash/
20 November, 2006
(at
18:39
)
Using stacks
Let
→
STKs
:=
(
Ξ
n
)
∞
n
=1
be
the
stacks
used
to
cut&stack a rank1
T
, on a nonatomic Lebesgue
space
(
X,
X
, μ
)
. Let
L
n
denote the height of Ξ
n
.
Let
S
n
⊂
X
denote the spacers adjoined to make
the
n
stack. Thus
Ξ
n
t
S
n
+1
= Ξ
n
+1
.
Let
A
n
:=
F
∞
j
=
n
+1
S
j
be the spacers adjoined a
fter
stage
n
. Certainly
μ
(
A
n
)
&
0.
I first construct a particular 2set generating parti
tion
P
=
(
B, G
)
.
Step 1.
For
(
Ξ
n
)
∞
n
=1
,
I will DTASARenumber
(
Drop To A Subsequence And Renumber
) several times,
so as to gain a new property for
→
STKs. Later subse
quencings will preserve the properties obtained ear
lier. I can initially DTASARenumber so that
L
n
>
n
+ 3, for all
n
.
First DTASARenumber so that there are at least
2
n
copies of Ξ
n
in Ξ
n
+1
. Now simply declare that the
bottommost copy of Ξ
n
in Ξ
n
+1
is, in fact, spacer
which is part of
S
n
+1
. Since
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 Fall '09
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 Negative and nonnegative numbers, ξN

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