We describe some of the interesting dynamical and topological properties of the complex exponential family λez and its associated Julia sets.

We consider two characteristic exponents of a rational function f:ℂ̂ → ℂ̂ of degree d ≥ 2. The exponent ${\chi }_a\left(f\right)$ is the average of log∥f’∥ with respect to the measure of maximal entropy. The exponent ${\chi }_m\left(f\right)$ can be defined as the maximal characteristic exponent over all periodic orbits of f. We prove that ${\chi }_a\left(f\right)={\chi }_m\left(f\right)$ if and only if f(z) is conformally conjugate to $z\mapsto z^{\pm d}$.