Talk at BU February 23, 2015.
Birkhoff sums over the golden rotation
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Oliver knill
We look at Birkhoff sums S_n(t)/n=sum_{k=1}^n X_k(t)/n with X_k(t)=g(T^kt)
where T is the irrational golden rotation and where g(t)=cot(pi t).
Similar sums have been studied by number theorists like Hardy and
Littlewood or Sinai and Ulcigrain in the context of the curlicue
problem. Birkhoff sums can be visualized if the time interval [0,n]
is rescaled so that it displays a graph over the interval [0,1]. While
for any L^1 function g(t) and ergodic T, the sum S_[n x](t)/n converges
a.e. to a linear function Mx by Birkhoff's ergodic theorem, there is
an interesting phenomenon for Cauchy distributed random variables like
if g(t) = cot(pi t) The function x -> S_[n x]/n on [0,1] converges
for n -> infinity to an explicitly given fractal limiting function,
if n is restricted to Fibonacci numbers F(2n) and if the start point t
is 0. The convergence to the ``golden graph" shows a truly self-similar
random walk. It explains some observations obtained together with John
Lesieutre and Folkert Tangerman, where we summed the anti derivative G of
g which is the Hilbert transform of a piecewise linear periodic function.
Birkhoff sums are relevant in KAM contexts, both in analytic and smooth
situations or in Denjoy-Koksma theory which is a refinement of Birkhoff
for Diophantine irrational rotations. In a probabilistic context, we have
a discrete time stochastic process modeling ``high risk" situations as
hitting a point near the singularity catastrophically changes the sum.
Diophantine conditions assure that there is enough time to ``recover"
from such a catastrophe. There are other connections like with modular
functions in number theory or Milnor's theorem telling that the cot
function is the unique non constant solution to the Kubert relation
(1/n) sum_{k=1}^n g(t+k/n) = g(t).
Earlier title version and abstract which did not make it:
Title: High risk: summing random variables of infinite variance
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Oliver Knill
We look at the growth of the Birkhoff sum S_n(t)/n=sum_{k=1}^n X_k(t)/n of
random variables X_k(t)=g(T^k t) where the transformation T:[0,1] ->[0,1]
is given by an irrational rotation and where the random variables X_k
have infinite variance as g is chosen to be outside of L^1[0,1]. This
is part of high risk probability theory. Such sums have been studied
early on by number theorists, like Hardy and Littlewood in the context
of the geometry of numbers. We observed a strange limiting behavior in
the case of the golden irrational rotation. Random walks and Birkhoff
sums can be visualized if the time interval [0,n] is rescaled so that
we see a graph over the interval [0,1]. For any L^1 function g(t) and
any ergodic transformation T, the Birkhoff sum S_[n x](t)/n converges to
a linear function Mx by the law of large numbers rsp Birkhoff's ergodic
theorem for almost all t. For non-integrable functions like g(t) = cot(pi
t) which was studied also by Sinai and Ulcigrai in the context of the
curlicue problem, something strange happens if the dynamical systems is
the golden rotation: the function x -> S_[n x]/n on [0,1] converges for
n -> infinity to an explicitly given fractal limiting function, if n is
restricted to Fibonnacci numbers F(2n) and if the start point t is 0. The
convergence to the ``golden graph" shows a truly self-similar random
walk. It explains some observations done with Lesieutre and Tangerman,
where we looked at the Birkhoff sum of the anti-derivative G of g which
is the Hilbert transform of a piecewise linear function whose Birkhoff
sum is studied in the geometry of numbers. There are other connections:
Birkhoff sums are relevant in KAM contexts both in analytic and smooth
situations or in Denjoy-Koksma theory which is a refinement of Birkhoff
for Diophantine irrational rotations. In a probabilistic context,
the distribution of X_k if g(t)=cot(pi t) is the Cauchy distribution
which plays a fundamental central role when summing up random variables
of infinite variance. Such discrete time stochastic processes can be
used to model ``high risk" situations, because hitting a point near the
singularity catastrophically changes the sum. Diophantine conditions
assure that there is enough time to ``recover" from such a catastrophe.
Important for central limits is the property that the sum of Cauchy
distributed random variables are again Cauchy. Related is the Milnor
theorem telling that the cot function is the unique non constant solution
to the Kubert renormalization relation (1/n) sum_{k=1}^n g(t+k/n) = g(t).
The Cauchy distribution is as ``central" among non L^1 random variables
as the Gaussian is among L^1 random variables. In solid state physics,
the same non-L^1 potentials have appeared as potentials in the Maryland
model in solid state physics leading to an explicitly known density of
states and localization in the Diophantine case. Finally, if the rotation
number alpha is complexified and chosen to be in the upper half plane,
the Birkhoff sum converges to an analytic function in alpha related to
the theta function and a modular form R=theta^2. This work relies on
papers written together with John Lesieutre and Folkert Tangerman.