Exponential map on the complex plane

Let f(z)=e^z=e^x(\cos y+\i\sin y) (for z=x+\i y) be the exponential map. Note that f(x)>0 for all real numbers and f^{n+1}(x):=f(f^nx) goes to \infty really fast: the dynamics of f on \mathbb{R} is trivial. But the dynamics of f on \mathbb{C} is completely different. First note that e^{2k\pi\i}=1: the map is not a diffeomorphism, but a covering map branching at the origin. The following theorem was conjectured by Fatou (1926) and proved by Misiurewicz (1981).

Theorem (Orbits of the complex exponential map).
Let \mathcal{O}_e(z) be the orbit of a point z\in\mathbb{C} under the iterates of f(z)=e^z. Then each of the following sets is dense in the complex plane:
1. the basin of \infty, B_e(\infty)=\{z\in\mathbb{C}: f^n(z)\to\infty\};
2. the set of transitive points, \text{Tran}(e)=\{z\in\mathbb{C}: \mathcal{O}_e(z)\text{ is dense}\};
3. the set of periodic points, \text{Per}(e)=\{z\in\mathbb{C}: \mathcal{O}_e(z)\text{ is finite}\}.

So the exponential map is chaotic on the complex plane.


The exponential map is chaotic: An invitation to transcendental dynamics,
Zhaiming Shen and Lasse Rempe-Gillen arXiv

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