Editing Complex dynamics (section)

===Lattès maps===
A '''[[Lattès map]]''' is an endomorphism ''f'' of <math>\mathbf{CP}^n</math> obtained from an endomorphism of an [[abelian variety]] by dividing by a [[finite group]]. In this case, the equilibrium measure of ''f'' is [[absolutely continuous measure|absolutely continuous]] with respect to [[Lebesgue measure]] on <math>\mathbf{CP}^n</math>. Conversely, by [[Anna Zdunik]], François Berteloot, and Christophe Dupont, the only endomorphisms of <math>\mathbf{CP}^n</math> whose equilibrium measure is absolutely continuous with respect to Lebesgue measure are the Lattès examples.<ref>Berteloot & Dupont (2005), Théorème 1.</ref> That is, for all non-Lattès endomorphisms, <math>\mu_f</math> assigns its full mass 1 to some [[Borel set]] of Lebesgue measure 0.
[[File:Equilibrium measure for Lattes map.png|thumb|A random sample from the equilibrium measure of the Lattès map <math>f(z)=(z-2)^2/z^2</math>. The Julia set is all of <math>\mathbf{CP}^1</math>.]]
[[File:Equilibrium measure for rational function.png|thumb|A random sample from the equilibrium measure of the non-Lattès map <math>f(z)=(z-2)^4/z^4</math>. The Julia set is all of <math>\mathbf{CP}^1</math>,<ref>Milnor (2006), problem 14-2.</ref> but the equilibrium measure is highly irregular.]]

In dimension 1, more is known about the "irregularity" of the equilibrium measure. Namely, define the ''Hausdorff dimension'' of a probability measure <math>\mu</math> on <math>\mathbf{CP}^1</math> (or more generally on a smooth manifold) by
:<math>\dim(\mu)=\inf \{\dim_H(Y):\mu(Y)=1\},</math>
where <math>\dim_H(Y)</math> denotes the Hausdorff dimension of a Borel set ''Y''. For an endomorphism ''f'' of <math>\mathbf{CP}^1</math> of degree greater than 1, Zdunik showed that the dimension of <math>\mu_f</math> is equal to the Hausdorff dimension of its support (the Julia set) if and only if ''f'' is conjugate to a Lattès map, a [[Chebyshev polynomial]] (up to sign), or a power map <math>f(z)=z^{\pm d}</math> with <math>d\geq 2</math>.<ref>Zdunik (1990), Theorem 2; Berteloot & Dupont (2005), introduction.</ref> (In the latter cases, the Julia set is all of <math>\mathbf{CP}^1</math>, a closed interval, or a circle, respectively.<ref>Milnor (2006), problem 5-3.</ref>) Thus, outside those special cases, the equilibrium measure is highly irregular, assigning positive mass to some closed subsets of the Julia set with smaller Hausdorff dimension than the whole Julia set.