Transcript The Relative Growth of Information Aimee S. A. Johnson Swarthmore College joint with Karma Dajani, Universiteit Utrecht Martijn de Vries, Technische Universiteit Delft.

The Relative Growth of
Information
Aimee S. A. Johnson
Swarthmore College
joint with
Karma Dajani, Universiteit Utrecht
Martijn de Vries, Technische Universiteit Delft
Scenario

Given
x  0,1

1  5
.x 2
Scenario


Given x   0,1
Express as

decimal expansion

1  5
.x 2

. x  .618033989...
Scenario


Given x   0,1
Express as



decimal expansion
continued fraction
expansion x 
a1 
1  5
.x 2


1
a2 
1
1
a3  ...
 [a1 , a2 ,...], ai  1
. x  .618033989...
. x  [1,1,1,...]
ai  1
Scenario


Given x   0,1
Express as



decimal expansion
continued fraction
expansion x 
a1 
Question:
1  5
. x
2


1
a2 
1
1
a3  ...
 [a1 , a2 ,...], ai  1
Given first n digits in decimal exp,
how many digits are known in
continued fraction expansion?
. x  .618033989...
.
x  [1,1,1,...]
,
Rephrase:
Given x starts with .d1d 2 ...d n
What is largest m=m(n,x) s.t.
we know x starts with [a1 , a2 ,..., am ( n , x ) ]?
Lochs’ Theorem:
,
Rephrase:
Given x starts with .d1d 2 ...d n
What is largest m=m(n,x) s.t.
we know x starts with [a1 , a2 ,..., am ( n , x ) ]?
Lochs’ Theorem: for a.e. x,
m(n, x)
n
Rephrase:
Given x starts with .d1d 2 ...d n
What is largest m=m(n,x) s.t.
we know x starts with [a1 , a2 ,..., am ( n , x ) ]?
Lochs’ Theorem:
m(n, x) 6 ln 2 ln10
Limn

n
2
Dynamical Systems Viewpoint
Bosma, Dajani, Kraaikamp
Dajani, Fieldsteel
Dynamical Systems Viewpoint
Bosma, Dajani, Kraaikamp
Dajani, Fieldsteel
Consider
([0,1), B   S)
Dynamical Systems Viewpoint
Bosma, Dajani, Kraaikamp
Dajani, Fieldsteel
Consider
([0,1), B   S)
B  Borel  -algebra
  Lebesgue measure
S x = 10x mod 1
Dynamical Systems Viewpoint
Bosma, Dajani, Kraaikamp
Dajani, Fieldsteel
Consider ([0,1), B   S)
B  Borel  -algebra
  Lebesgue measure
S x = 10x mod 1

Partition P
Dynamical Systems Viewpoint
Bosma, Dajani, Kraaikamp
Dajani, Fieldsteel
Consider ([0,1), B   S)
B  Borel  -algebra
  Lebesgue measure
S x = 10x mod 1

Partition P
p0 p1 p 2 p3 p 4 p5 p 6 p 7 p8 p9
._._._._._._._._._._.
0
.5
1
Dynamical Systems Viewpoint
Bosma, Dajani, Kraaikamp
Dajani, Fieldsteel
Consider ([0,1), B   S)
B  Borel  -algebra
Partition P
0 1 2 3 4 5 6 7 8 9
p
p p p p p p p p p
._._._._._._._._._._.
  Lebesgue measure
0
S x = 10x mod 1
Or
([0,1), B   T )
  Gauss measure
Tx
1
mod 1
x

.5
1
Dynamical Systems Viewpoint
Bosma, Dajani, Kraaikamp
Dajani, Fieldsteel
([0,1), B   S)
B  Borel  -algebra
Partition P
0 1 2 3 4 5 6 7 8 9
p
p p p p p p p p p
._._._._._._._._._._.
  Lebesgue measure
0
Consider

.5
1
S x = 10x mod 1
Or
([0,1), B   T )
  Gauss measure
1
Tx
mod 1
x
Partition Q
1
q
q q q
.__._._.__.______.
4
3
2
0 .2 .25 .33 .5
1
1
P

P

S
P
Let n
e.g.
and
 S n P
P1  {[0,.01),[.01,.02),....[.99,1)}
pn ( x)  element of Pn that x lies in
(tells you part of itinerary of x)
Same for
Question:
Qm , qm ( x)
Given pn ( x),
what is largest m=m(n, x) s.t.
pn ( x)  qm ( x)?
Tools

Generating partition
a.e. x≠y, there exists n s.t pn ( x)  pn ( y).

Ergodic transformations
S 1 ( A)  A   ( A)  0 or 1

Entropy
nonnegative number which indicates amount of
uncertainty in system = hλ(S)

Shannon-McMillan-Breiman Theorem
For T ergodic, P generating, a.e. x ,
Lim n
log  ( pn ( x))
 h (T )   (p n ( x))
n
2  nh (T )
Theorem
Given 2 ergodic dynamical systems on [0,1)
with generating interval partitions P and Q,
with entropies c and d,
for a.e. x,
Lim n 
m(n, x) c

n
d
Theorem
Given 2 ergodic dynamical systems on [0,1)
with generating interval partitions P and Q,
with entropies c and d,
for a.e. x,
Lim n 
e.g.
m(n, x) c

n
d
h ( S )  ln10  c
h (T ) 
2
6 ln 2
d
Higher Dimensions


.x  [0,1) 2
Assumptions
 . Pn  squares
 . Qm  convex polygons
 . r  ( q )  (diam q )   s  ( q )
m
m
m
Theorem

Given 2 ergodic dynamical systems on
with generating partitions P and Q
with entropies c and d,
Then for a.e. x,
Limn 
m(n, x)

c

.
n
2(   1) d
[0,1) 2
Vague idea of proof
.
Vague idea of proof

c
n
2(   1) d
Let M=
When wouldn’t m(n,x) = M?
Vague idea of proof

c
n
2(   1) d
Let M=
When wouldn’t m(n,x) = M?
bad points; where pn ( x)  qM ( x)
where x in is “frame” of
qM ( x)
Vague idea of proof

c
n
2(   1) d
Let M=
When wouldn’t m(n,x) = M?
bad points; where pn ( x)  qM ( x)
where x in is “frame” of
qM ( x)
So msr of bad pts ≤ msr of frames
 n
2
≈k
Vague idea of proof

c
n
2(   1) d
Let M=
When wouldn’t m(n,x) = M?
bad points; where pn ( x)  qM ( x)
where x in is “frame” of
qM ( x)
So msr of bad pts ≤ msr of frames
 n
2
≈k
So Σ(bad pts at nth stage) < 
So a.e. x leaves bad set eventually
With thanks to the organizers of the MSRI
Connections for Women, January 2007