↑ arvau

Wikipedia / Complex dynamics

Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by iterating a complex analytic mapping. This article focuses on the case of algebraic dynamics, where a polynomial or rational function is iterated. In geometric terms, that amounts to iterating a mapping from some algebraic variety to itself. The related theory of arithmetic dynamics studies iteration over the rational numbers or the p-adic numbers instead of the complex numbers.

Dynamics in complex dimension 1

from the complex numbers C to itself. It is helpful to view this as a map from the complex projective line has the advantage of being compact, how does its orbit (or forward orbit)

behave, qualitatively? The answer is: if the absolute value |z| is less than 1, then the orbit converges to 0, in fact more than exponentially fast. If |z are superattracting fixed pointss of f, meaning that the derivative of f is zero at those points. An attracting fixed point means one where the derivative of f has absolute value less than 1.)

, meaning that z is on the unit circle in C. At these points, the dynamics of f is chaotic, in various ways. For example, for almost all points z on the circle in terms of measure theory, the forward orbit of z is dense in the circle, and in fact uniformly distributed on the circle. There are also infinitely many periodic points for some positive integer r means the result of applying f to z r.) Even at periodic points z on the circle, the dynamics of f can be considered chaotic, since points near z diverge exponentially fast from z upon iterating f. (The periodic points of f on the unit circle are repelling at z has absolute value greater than 1.)

Pierre Fatou and Gaston Julia to itself of degree, the Julia set, on which the dynamics of f, the Julia set is the unit circle. For other polynomial mappings, the Julia set is often highly irregular, for example a fractal in the sense that its Hausdorff dimension. The Mandelbrot set is the set of complex numbers c is connected.

There is a rather complete classification of the possible dynamics in the Fatou set, the complement of the Julia set, where the dynamics is "tame". Namely, Dennis Sullivan showed that each connected component U. Therefore, to analyze the dynamics on a component U, one can assume after replacing f. 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. (Note that the "backward orbit" of a point z in U that map to z under some iterate of f, need not be contained in U.)

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 have been extended to a class of rational maps from any projective variety to itself. Note, however, that many varieties have no interesting self-maps.

Let f, meaning a morphism of algebraic varieties to itself, for a positive integer n. Such a mapping is given in homogeneous coordinates by

of the same degree d. (By Chow's theorem, this is the same thing as a holomorphic to itself.) Assume that d is greater than 1; then the degree of the mapping f, which is also greater than 1.

Then there is a unique probability measure, 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 Lopes, Ricardo Mañé, and Mikhail Lyubich (around 1983), and by John Hubbard, Peter Papadopol, John Fornaess, and Nessim Sibony in any dimension (around 1994). The small Julia set is the support.

Examples

  • is the Haar measure.

  • be the mapping

  • is the Haar measure on the n-dimensional torus 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. Because f is a finite morphism is totally invariant.

when followed backward in time, by Jean-Yves Briend, Julien Duval, Tien-Cuong Dinh, and Sibony. Namely, for a point z and a positive integer r points w. Then there is a Zariski closed such that for all points z not in E, the measures just defined converge weakly as r are totally invariant under f), and one can take the exceptional set E.

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. Consider the probability measure which is evenly distributed on the points of period r as r..

that is not the whole space., it follows that the periodic points of f are Zariski dense. A more algebraic proof of this Zariski density was given by Najmuddin Fakhruddin. has no isolated points, and so it is a perfect set.

of the equilibrium measure is not too small, in the sense that its Hausdorff dimension is always greater than zero..) Another way to make precise that f has some chaotic behavior is that the topological entropy of f, by Mikhail Gromov, Michał Misiurewicz, and Feliks Przytycki.

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 is the unique invariant measure of maximal entropy, by Briend and Duval. 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 with respect to that measure, by Fornaess and Sibony..

Lattès maps

A Lattès map is an endomorphism f 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 with respect to Lebesgue measure. Conversely, by Anna Zdunik whose equilibrium measure is absolutely continuous with respect to Lebesgue measure are the Lattès examples. assigns its full mass 1 to some Borel set of Lebesgue measure 0.

In dimension 1, more is known about the "irregularity" of the equilibrium measure. Namely, define the Hausdorff dimension (or more generally on a smooth manifold) by

denotes the Hausdorff dimension of a Borel set Y. For an endomorphism f 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., a closed interval, or a circle, respectively.) 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.

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 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.

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. Explicitly, for X of complex dimension n be the spectral radius of f acting by pullback on the Hodge cohomology. Then the topological entropy of f is

(The topological entropy of f is also the logarithm of the spectral radius of f.) 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 surfaces and K3 surfaces do have such automorphisms.

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 takes its maximum value, the action of f, 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. (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 of maximal entropy for f, called the equilibrium measure (or Green measure, or measure of maximal entropy). with respect to f is called the small Julia set. Informally: f has some chaotic behavior, and the most chaotic behavior is concentrated on the small Julia set. At least when X assigns zero mass to all sets of sufficiently small Hausdorff dimension.)

Kummer automorphisms

Some abelian varieties have an automorphism of positive entropy. For example, let E be a complex elliptic curve and let X integer matrices acts on X. Any group element f whose trace, 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.

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 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. 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 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. vanishes on closed complex subspaces not equal to X. It follows that the periodic points of f) are Zariski dense in X.

For an automorphism f with simple action on cohomology, f. It follows that for almost every point z, the forward and backward orbits of z.

is that for an automorphism f with simple action on cohomology, there can be a nonempty open subset of X 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. 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. Consider the probability measure which is evenly distributed on the isolated periodic points of period r as r goes to infinity, by Eric Bedford, Lyubich, and John Smillie..

See also

Dynamics in complex dimension 1

Related areas of dynamics

Notes

References