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

The family of exponential functions is called the exponential family.

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.

${\displaystyle Z=x+y*i}$

${\displaystyle \exp(Z)=e^{Z}}$

${\displaystyle \mathrm {Real(\exp(Z))} =\exp(x)\cos(y)}$ 

${\displaystyle \mathrm {Imag(\exp(Z))} =\exp(x)\sin(y)}$

"The function

 ${\displaystyle e^{x}-1}$ 

is one of the simpler applications of continuous iteration. The reason why is because regular iteration requires a fixed point in order to work, and this function has a very simple fixed point, namely zero: "^[3]

 ${\displaystyle e^{0}-1=0}$

• commons Category:Exponential maps

• Julia_and_Mandelbrot_sets_for_transcendental_functions by Gertbuschmann
• Exponential mapping of the plane
• Exponential maps ^[4][5][6]

^[7]

• tetration fractals ^[8]
• Baker domains ^[9][10]

1. ↑ Dynamics of exponential maps by Lasse Rempe
2. ↑ "Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity", Lasse Rempe, Dierk Schleicher
3. ↑ Tetration FAQ by Henryk Trappman Andrew Robbins July 10, 2008
4. ↑ THE EXPONENTIAL MAP IS CHAOTIC: AN INVITATION TO TRANSCENDENTAL DYNAMICS by ZHAIMING SHEN AND LASSE REMPE-GILLEN
5. ↑ Dynamics of exponential maps by Lasse Rempe
6. ↑ wikipedia : Exponential map (discrete dynamical systems)
7. ↑ Paper by N Fagella
8. ↑ Paul Bourke fractals tetration
9. ↑ On the Stability of Julia Sets of Functions having Baker Domains by Arnd Lauber ( 2004)
10. ↑ Approximation of Baker domains and convergence of Julia sets by Tania Garfias-Macedo aus Mexiko Stadt, Mexiko