To illustrate an application to our Birkhoff sum, we
introduced the following map in the two complex dimensional space C^{2}
(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|w_{n}| is equal to S_{n}/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. |