Birkhoff sum project 2010
A quadratic map in two complex dimensionalal space
The quadratic map T(z,w)=(cz,w-wz)
Office: SciCtr 434
Email: knill@math.harvard.edu
To illustrate an application to our Birkhoff sum, we
introduced the following map in the two complex dimensional space C2
(z,w) = (c z, w-w z)
where c=exp(2Pi i phi) is a constant complex number and where phi is the
golden ratio. It is one of the simplest nonlinear maps in the complex plane
one can think of. It does not have a interesting analogue in the real because
|c|=1 is necessary to make the dynamics nontrivial.
All the nontrivial dynamics of this map happens on |z|=1
where the log|wn| is equal to Sn/2 with our Birkhoff
sum. For |z| larger than 1, the second coordinate explodes to infinity. For
r=|z| smaller than one is understood by Gotttschalk Hedlund: the
orbit is on a circle c(t) = (exp(i t), A(t)) where A(t) in C is a circle.
How does this attractor look like when r is changed from 0 to 1?
Animation.
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