Let (for ) be the exponential map. Note that for all real numbers and goes to really fast: the dynamics of on is trivial. But the dynamics of on is completely different. First note that : 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 be the orbit of a point under the iterates of . Then each of the following sets is dense in the complex plane:

1. the basin of , ;

2. the set of transitive points, ;

3. the set of periodic points, .

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