Editing Complex dynamics (section)

==Automorphisms of projective varieties==
More generally, complex dynamics seeks to describe the behavior of rational maps under iteration. One case that has been studied with some success is that of ''automorphisms'' of a [[smooth scheme|smooth]] complex projective variety ''X'', meaning isomorphisms ''f'' from ''X'' to itself. The case of main interest is where ''f'' acts nontrivially on the [[singular cohomology]] <math>H^*(X,\mathbf{Z})</math>.

Gromov and Yosef Yomdin showed that the topological entropy of an endomorphism (for example, an automorphism) of a smooth complex projective variety is determined by its action on cohomology.<ref>Cantat (2000), Théorème 2.2.</ref> Explicitly, for ''X'' of complex dimension ''n'' and <math>0\leq p\leq n</math>, let <math>d_p</math> be the [[spectral radius]] of ''f'' acting by pullback on the [[Hodge theory|Hodge cohomology]] group <math>H^{p,p}(X)\subset H^{2p}(X,\mathbf{C})</math>. Then the topological entropy of ''f'' is
:<math>h(f)=\max_p \log d_p.</math>
(The topological entropy of ''f'' is also the logarithm of the spectral radius of ''f'' on the whole cohomology <math>H^*(X,\mathbf{C})</math>.) Thus ''f'' has some chaotic behavior, in the sense that its topological entropy is greater than zero, if and only if it acts on some cohomology group with an [[eigenvalue]] of absolute value greater than 1. Many projective varieties do not have such automorphisms, but (for example) many [[rational surface]]s and [[K3 surface]]s do have such automorphisms.<ref>Cantat (2010), sections 7 to 9.</ref>

Let ''X'' be a compact [[Kähler manifold]], which includes the case of a smooth complex projective variety. Say that an automorphism ''f'' of ''X'' has ''simple action on cohomology'' if: there is only one number ''p'' such that <math>d_p</math> takes its maximum value, the action of ''f'' on <math>H^{p,p}(X)</math> has only one eigenvalue with absolute value <math>d_p</math>, and this is a [[simple eigenvalue]]. For example, [[Serge Cantat]] showed that every automorphism of a compact Kähler surface with positive topological entropy has simple action on cohomology.<ref>Cantat (2014), section 2.4.3.</ref> (Here an "automorphism" is complex analytic but is not assumed to preserve a Kähler metric on ''X''. In fact, every automorphism that preserves a metric has topological entropy zero.)

For an automorphism ''f'' with simple action on cohomology, some of the goals of complex dynamics have been achieved. Dinh, Sibony, and Henry de Thélin showed that there is a unique invariant probability measure <math>\mu_f</math> of maximal entropy for ''f'', called the '''equilibrium measure''' (or '''Green measure''', or '''measure of maximal entropy''').<ref>De Thélin & Dinh (2012), Theorem 1.2.</ref> (In particular, <math>\mu_f</math> has entropy <math>\log d_p</math> with respect to ''f''.) The support of <math>\mu_f</math> is called the '''small Julia set''' <math>J^*(f)</math>. Informally: ''f'' has some chaotic behavior, and the most chaotic behavior is concentrated on the small Julia set. At least when ''X'' is projective, <math>J^*(f)</math> has positive Hausdorff dimension. (More precisely, <math>\mu_f</math> assigns zero mass to all sets of sufficiently small Hausdorff dimension.)<ref name="super">Dinh & Sibony (2010), "Super-potentials ...", section 4.4.</ref>

===Kummer automorphisms===
Some abelian varieties have an automorphism of positive entropy. For example, let ''E'' be a complex [[elliptic curve]] and let ''X'' be the abelian surface <math>E\times E</math>. Then the group <math>GL(2,\mathbf{Z})</math> of invertible <math>2\times 2</math> integer matrices acts on ''X''. Any group element ''f'' whose [[trace (linear algebra)|trace]] has absolute value greater than 2, for example <math>\begin{pmatrix}2&1\\1&1\end{pmatrix}</math>, has spectral radius greater than 1, and so it gives a positive-entropy automorphism of ''X''. The equilibrium measure of ''f'' is the Haar measure (the standard Lebesgue measure) on ''X''.<ref>Cantat & Dupont (2020), section 1.2.1.</ref>

The '''Kummer automorphisms''' are defined by taking the quotient space by a finite group of an abelian surface with automorphism, and then [[blowing up]] to make the surface smooth. The resulting surfaces include some special K3 surfaces and rational surfaces. For the Kummer automorphisms, the equilibrium measure has support equal to ''X'' and is [[smooth function|smooth]] outside finitely many curves. Conversely, Cantat and Dupont showed that for all surface automorphisms of positive entropy except the Kummer examples, the equilibrium measure is not absolutely continuous with respect to Lebesgue measure.<ref>Cantat & Dupont (2020), Main Theorem.</ref> In this sense, it is usual for the equilibrium measure of an automorphism to be somewhat irregular.

===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>.