Editing Complex dynamics (section)

===Saddle periodic points===
A periodic point ''z'' of ''f'' is called a ''saddle'' periodic point if, for a positive integer ''r'' such that <math>f^r(z)=z</math>, at least one eigenvalue of the derivative of <math>f^r</math> on the tangent space at ''z'' has absolute value less than 1, at least one has absolute value greater than 1, and none has absolute value equal to 1. (Thus ''f'' is expanding in some directions and contracting at others, near ''z''.) For an automorphism ''f'' with simple action on cohomology, the saddle periodic points are dense in the support <math>J^*(f)</math> of the equilibrium measure <math>\mu_f</math>.<ref name="super" /> On the other hand, the measure <math>\mu_f</math> vanishes on closed complex subspaces not equal to ''X''.<ref name="super" /> It follows that the periodic points of ''f'' (or even just the saddle periodic points contained in the support of <math>\mu_f</math>) are Zariski dense in ''X''.

For an automorphism ''f'' with simple action on cohomology, ''f'' and its inverse map are ergodic and, more strongly, mixing with respect to the equilibrium measure <math>\mu_f</math>.<ref>Dinh & Sibony (2010), "Super-potentials ...", Theorem 4.4.2.</ref> It follows that for almost every point ''z'' with respect to <math>\mu_f</math>, the forward and backward orbits of ''z'' are both uniformly distributed with respect to <math>\mu_f</math>.

A notable difference with the case of endomorphisms of <math>\mathbf{CP}^n</math> is that for an automorphism ''f'' with simple action on cohomology, there can be a nonempty open subset of ''X'' on which neither forward nor backward orbits approach the support <math>J^*(f)</math> of the equilibrium measure. For example, Eric Bedford, Kyounghee Kim, and [[Curtis McMullen]] constructed automorphisms ''f'' of a smooth projective rational surface with positive topological entropy (hence simple action on cohomology) such that ''f'' has a Siegel disk, on which the action of ''f'' is conjugate to an irrational rotation.<ref>Cantat (2010), Théorème 9.8.</ref> Points in that open set never approach <math>J^*(f)</math> under the action of ''f'' or its inverse.

At least in complex dimension 2, the equilibrium measure of ''f'' describes the distribution of the isolated periodic points of ''f''. (There may also be complex curves fixed by ''f'' or an iterate, which are ignored here.) Namely, let ''f'' be an automorphism of a compact Kähler surface ''X'' with positive topological entropy <math>h(f)=\log d_1</math>. Consider the probability measure which is evenly distributed on the isolated periodic points of period ''r'' (meaning that <math>f^r(z)=z</math>). Then this measure converges weakly to <math>\mu_f</math> as ''r'' goes to infinity, by Eric Bedford, Lyubich, and [[John Smillie (mathematician)|John Smillie]].<ref>Cantat (2014), Theorem 8.2.</ref> The same holds for the subset of saddle periodic points, because both sets of periodic points grow at a rate of <math>(d_1)^r</math>.