Editing Complex dynamics (section)

===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>.