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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 $\pm \frac{\pi}{2}$.

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  • $\begingroup$ Complex dynamics for non-holomorphic functions is not complex dynamics, it's just dynamics! Holomorphic dynamics is just a small part of the study of dynamical systems. In your case, you have a real polynomial map from $\mathbb R^2$ to itself, and it is dubious that writing it as a complex function will help. You can still make computer simulations fairly easily, using e.g. scilab (I'm not aware of any convenient software for that purpose but I'm sure it exists). $\endgroup$ – Glougloubarbaki Feb 20 at 15:15
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Maybe we should consider $N(z)=u(x,y)+i v(x,y)$ and further analyze the real-valued functions

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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).

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  • $\begingroup$ I am not at all well-versed in algebraic geometry, and am a little rusty even on multivariable cal. Do you have any resources that might help me to brush up on the concepts you cite? Thanks! $\endgroup$ – matty_k_walrus Feb 20 at 21:33
  • $\begingroup$ @matty_k_walrus this is not algebraic geometry, it's general dynamical systems theory. you can read eg en.wikipedia.org/wiki/Stable_manifold_theorem and the relevant links inside the page $\endgroup$ – Glougloubarbaki Feb 21 at 9:35
  • $\begingroup$ the idea is the near $(0,0)$, your map will have the same dynamics as its linear part, ie $(x,y) \mapsto (\frac{11}{9}x,\frac{7}{9}y)$ $\endgroup$ – Glougloubarbaki Feb 21 at 9:36

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