Complex dynamics for non-holomorphic functions

Question

Are there any resources for studying the dynamics of functions that are not holomorphic? Specifically, I am trying to study (for a paper) the behaviour of the function $$N(z) := z - \frac{2\overline{z}\left(z^2 - 1\right)}{9}$$ especially when its seeds have small moduli and have arguments near $\pm \frac{\pi}{2}$.

Answer 1

Maybe we should consider $N(z)=u(x,y)+i v(x,y)$ and further analyze the real-valued functions

Answer 2

Note that your function has a fixed point at $z=(x,y)=(0,0)$. Using real coordinates, we have $f(x,y)=(\frac{11}{9}x,\frac{7}{9}y)+h.o.t.$, so the origin is a saddle fixed point, with stable manifold tangent to the vertical axis. Therefore, for the values you mention, the dynamics should move points towards the origin for some time. If you start exactly on the stable manifold, you will converge to $(0,0)$ at exponential rate $\frac{7}{9}$. If you are not exactly on the stable manifold, you will come close to the origin then start getting away.

Analyzing the long term behaviour of points not on the stable manifold is a priori much more complicated (and possibly hopeless).