In the theory ofdynamical systems, theexponential map can be used as theevolution function ofthe discrete nonlinear dynamical system.^[1]

The family ofexponential functions is called theexponential family.

There are manyforms 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.

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