# Fractals/exponential

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

## Family

[edit | edit source]The family of exponential functions is called the **exponential family**.

## Forms

[edit | edit source]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:

The second one can be mapped to the first using the fact that , so is the same under the transformation . 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.

# How to compute it

[edit | edit source]```
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# What is the continous iteration of ?

[edit | edit source]"The function

```
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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]}

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# Images

[edit | edit source]# See also

[edit | edit source]- 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]}

# References

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