# Exponential map (discrete dynamical systems)

Parameter plane of the complex exponential family f(z)=exp(z)+c with 8 external ( parameter) rays

In the theory of dynamical systems, the exponential map can be used as the evolution function of the discrete nonlinear dynamical system.[1]

## Family

The family of exponential functions is called the exponential family.

## Forms

There are many forms of these maps,[2] many of which are equivalent under a coordinate transformation. For example two of the most common ones are:

• ${\displaystyle E_{c}:z\to e^{z}+c}$
• ${\displaystyle E_{\lambda }:z\to \lambda *e^{z}}$

The second one can be mapped to the first using the fact that ${\displaystyle \lambda *e^{z}.=e^{z+ln(\lambda )}}$, so ${\displaystyle E_{\lambda }:z\to e^{z}+ln(\lambda )}$ is the same under the transformation ${\displaystyle z=z+ln(\lambda )}$. The only difference is that, due to multi-valued properties of exponentiation, there may be a few select cases that can only be found in one version. Similar arguments can be made for many other formulas.

## References

1. ^ Dynamics of exponential maps by Lasse Rempe
2. ^ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity