Complex dynamics

This article focuses on the case of algebraic dynamics, where a polynomial or rational function is iterated.

A simple example that shows some of the main issues in complex dynamics is the mapping

Pierre Fatou and Gaston Julia showed in the late 1910s that much of this story extends to any complex algebraic map from

For other polynomial mappings, the Julia set is often highly irregular, for example a fractal in the sense that its Hausdorff dimension is not an integer.

Namely, Dennis Sullivan showed that each connected component U of the Fatou set is pre-periodic, meaning that there are natural numbers

Then either (1) U contains an attracting fixed point for f; (2) U is parabolic in the sense that all points in U approach a fixed point in the boundary of U; (3) U is a Siegel disk, meaning that the action of f on U is conjugate to an irrational rotation of the open unit disk; or (4) U is a Herman ring, meaning that the action of f on U is conjugate to an irrational rotation of an open annulus.

This measure was defined by Hans Brolin (1965) for polynomials in one variable, by Alexandre Freire, Artur Lopes, Ricardo Mañé, and Mikhail Lyubich for

One striking characterization of the equilibrium measure is that it describes the asymptotics of almost every point in

when followed backward in time, by Jean-Yves Briend, Julien Duval, Tien-Cuong Dinh, and Sibony.

), and one can take the exceptional set E to be the unique largest totally invariant closed complex subspace not equal to

[6] The equilibrium measure gives zero mass to any closed complex subspace of

of the equilibrium measure is not too small, in the sense that its Hausdorff dimension is always greater than zero.

[9]) 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

Conversely, by Anna Zdunik, François Berteloot, and Christophe Dupont, the only endomorphisms of

[15]) 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.

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 complex projective variety X, meaning isomorphisms f from X to itself.

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.

[17] Let X be a compact Kähler manifold, which includes the case of a smooth complex projective variety.

For example, Serge Cantat showed that every automorphism of a compact Kähler surface with positive topological entropy has simple action on cohomology.

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

assigns zero mass to all sets of sufficiently small Hausdorff dimension.

For the Kummer automorphisms, the equilibrium measure has support equal to X and is 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.

For an automorphism f with simple action on cohomology, the saddle periodic points are dense in the support

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

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

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.

Namely, let f be an automorphism of a compact Kähler surface X with positive topological entropy

The Julia set of the polynomial with .
The Julia set of the polynomial with . This is a Cantor set .
A random sample from the equilibrium measure of the Lattès map . The Julia set is all of .
A random sample from the equilibrium measure of the non-Lattès map . The Julia set is all of , [ 13 ] but the equilibrium measure is highly irregular.