Editing Complex dynamics (section)

==The equilibrium measure of an endomorphism==
Complex dynamics has been effectively developed in any dimension. This section focuses on the mappings from [[complex projective space]] <math>\mathbf{CP}^n</math> to itself, the richest source of examples. The main results for <math>\mathbf{CP}^n</math> have been extended to a class of [[rational map]]s from any [[projective variety]] to itself.<ref>Guedj (2010), Theorem B.</ref> Note, however, that many varieties have no interesting self-maps.

Let ''f'' be an endomorphism of <math>\mathbf{CP}^n</math>, meaning a [[morphism of algebraic varieties]] from <math>\mathbf{CP}^n</math> to itself, for a positive integer ''n''. Such a mapping is given in [[homogeneous coordinates]] by
:<math>f([z_0,\ldots,z_n])=[f_0(z_0,\ldots,z_n),\ldots,f_n(z_0,\ldots,z_n)]</math>
for some homogeneous polynomials <math>f_0,\ldots,f_n</math> of the same degree ''d'' that have no common zeros in <math>\mathbf{CP}^n</math>. (By [[Algebraic_geometry_and_analytic_geometry#Chow's_theorem|Chow's theorem]], this is the same thing as a [[holomorphic]] mapping from <math>\mathbf{CP}^n</math> to itself.) Assume that ''d'' is greater than 1; then the degree of the mapping ''f'' is <math>d^n</math>, which is also greater than 1.

Then there is a unique [[probability measure]] <math>\mu_f</math> on <math>\mathbf{CP}^n</math>, the '''equilibrium measure''' of ''f'', that describes the most chaotic part of the dynamics of ''f''. (It has also been called the '''Green measure''' or '''measure of maximal entropy'''.) This measure was defined by Hans Brolin (1965) for polynomials in one variable, by Alexandre Freire, [[Artur Oscar Lopes|Artur Lopes]], [[Ricardo Mañé]], and [[Mikhail Lyubich]] for <math>n=1</math> (around 1983), and by [[John H. Hubbard|John Hubbard]], Peter Papadopol, [[John Fornaess]], and [[Nessim Sibony]] in any dimension (around 1994).<ref name="measure">Dinh & Sibony (2010), "Dynamics ...", Theorem 1.7.11.</ref> The '''small Julia set''' <math>J^*(f)</math> is the [[support (measure theory)|support]] of the equilibrium measure in <math>\mathbf{CP}^n</math>; this is simply the Julia set when <math>n=1</math>.

===Examples===
* For the mapping <math>f(z)=z^2</math> on <math>\mathbf{CP}^1</math>, the equilibrium measure <math>\mu_f</math> is the [[Haar measure]] (the standard measure, scaled to have total measure 1) on the unit circle <math>|z|=1</math>.
* More generally, for an integer <math>d>1</math>, let <math>f\colon \mathbf{CP}^n\to\mathbf{CP}^n</math> be the mapping
::<math>f([z_0,\ldots,z_n])=[z_0^d,\ldots,z_n^d].</math>
:Then the equilibrium measure <math>\mu_f</math> is the Haar measure on the ''n''-dimensional [[torus]] <math>\{[1,z_1,\ldots,z_n]: |z_1|=\cdots=|z_n|=1\}.</math> For more general holomorphic mappings from <math>\mathbf{CP}^n</math> to itself, the equilibrium measure can be much more complicated, as one sees already in complex dimension 1 from pictures of Julia sets.

===Characterizations of the equilibrium measure===
A basic property of the equilibrium measure is that it is ''invariant'' under ''f'', in the sense that the [[pushforward measure]] <math>f_*\mu_f</math> is equal to <math>\mu_f</math>. Because ''f'' is a [[finite morphism]], the pullback measure <math>f^*\mu_f</math> is also defined, and <math>\mu_f</math> is '''totally invariant''' in the sense that <math>f^*\mu_f=\deg(f)\mu_f</math>.

One striking characterization of the equilibrium measure is that it describes the asymptotics of almost every point in <math>\mathbf{CP}^n</math> when followed backward in time, by Jean-Yves Briend, Julien Duval, [[Dinh Tien-Cuong|Tien-Cuong Dinh]], and Sibony. Namely, for a point ''z'' in <math>\mathbf{CP}^n</math> and a positive integer ''r'', consider the probability measure <math>(1/d^{rn})(f^r)^*(\delta_z)</math> which is evenly distributed on the <math>d^{rn}</math> points ''w'' with <math>f^r(w)=z</math>. Then there is a [[Zariski closed]] subset <math>E\subsetneq \mathbf{CP}^n</math> such that for all points ''z'' not in ''E'', the measures just defined [[weak convergence of measures|converge weakly]] to the equilibrium measure <math>\mu_f</math> as ''r'' goes to infinity. In more detail: only finitely many closed complex subspaces of <math>\mathbf{CP}^n</math> are '''totally invariant''' under ''f'' (meaning that <math>f^{-1}(S)=S</math>), and one can take the ''exceptional set'' ''E'' to be the unique largest totally invariant closed complex subspace not equal to <math>\mathbf{CP}^n</math>.<ref>Dinh & Sibony (2010), "Dynamics ...", Theorem 1.4.1.</ref>

Another characterization of the equilibrium measure (due to Briend and Duval) is as follows. For each positive integer ''r'', the number of periodic points of period ''r'' (meaning that <math>f^r(z)=z</math>), counted with multiplicity, is <math>(d^{r(n+1)}-1)/(d^r-1)</math>, which is roughly <math>d^{rn}</math>. Consider the probability measure which is evenly distributed on the points of period ''r''. Then these measures also converge to the equilibrium measure <math>\mu_f</math> as ''r'' goes to infinity. Moreover, most periodic points are repelling and lie in <math>J^*(f)</math>, and so one gets the same limit measure by averaging only over the repelling periodic points in <math>J^*(f)</math>.<ref>Dinh & Sibony (2010), "Dynamics ...", Theorem 1.4.13.</ref> There may also be repelling periodic points outside <math>J^*(f)</math>.<ref>Fornaess & Sibony (2001), Theorem 4.3.</ref>

The equilibrium measure gives zero mass to any closed complex subspace of <math>\mathbf{CP}^n</math> that is not the whole space.<ref name="subspace">Dinh & Sibony (2010), "Dynamics ...", Proposition 1.2.3.</ref> Since the periodic points in <math>J^*(f)</math> are dense in <math>J^*(f)</math>, it follows that the periodic points of ''f'' are [[Zariski dense]] in <math>\mathbf{CP}^n</math>. A more algebraic proof of this Zariski density was given by Najmuddin Fakhruddin.<ref>Fakhruddin (2003), Corollary 5.3.</ref> Another consequence of <math>\mu_f</math> giving zero mass to closed complex subspaces not equal to <math>\mathbf{CP}^n</math> is that each point has zero mass. As a result, the support <math>J^*(f)</math> of <math>\mu_f</math> has no isolated points, and so it is a [[perfect set]].

The support <math>J^*(f)</math> of the equilibrium measure is not too small, in the sense that its Hausdorff dimension is always greater than zero.<ref name="subspace" /> In that sense, an endomorphism of complex projective space with degree greater than 1 always behaves chaotically at least on part of the space. (There are examples where <math>J^*(f)</math> is all of <math>\mathbf{CP}^n</math>.<ref>Milnor (2006), Theorem 5.2 and problem 14-2; Fornaess (1996), Chapter 3.</ref>) Another way to make precise that ''f'' has some chaotic behavior is that the [[topological entropy]] of ''f'' is always greater than zero, in fact equal to <math>n\log d</math>, by [[Mikhail Gromov (mathematician)|Mikhail Gromov]], [[Michał Misiurewicz]], and Feliks Przytycki.<ref>Dinh & Sibony (2010), "Dynamics ...", Theorem 1.7.1.</ref> 

For any continuous endomorphism ''f'' of a compact metric space ''X'', the topological entropy of ''f'' is equal to the maximum of the [[measure-theoretic entropy]] (or "metric entropy") of all ''f''-invariant measures on ''X''. For a holomorphic endomorphism ''f'' of <math>\mathbf{CP}^n</math>, the equilibrium measure <math>\mu_f</math> is the ''unique'' invariant measure of maximal entropy, by Briend and Duval.<ref name="measure" /> This is another way to say that the most chaotic behavior of ''f'' is concentrated on the support of the equilibrium measure.

Finally, one can say more about the dynamics of ''f'' on the support of the equilibrium measure: ''f'' is [[ergodic]] and, more strongly, [[mixing (mathematics)|mixing]] with respect to that measure, by Fornaess and Sibony.<ref>Dinh & Sibony (2010), "Dynamics ...", Theorem 1.6.3.</ref> It follows, for example, that for almost every point with respect to <math>\mu_f</math>, its forward orbit is uniformly distributed with respect to <math>\mu_f</math>.

===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.