## 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.

Reference:

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