Complex dynamics

 ( Complex Analysis ) 

Complex dynamics, or holomorphic dynamics, is the study of 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 or the instead of the .

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Dynamics in complex dimension 1 A simple example that shows some of the main issues in complex dynamics is the mapping $f(z)=z^2$ from the complex numbers C to itself. It is helpful to view this as a map from the complex projective line $\backslash mathbf\{CP\}^1$ to itself, by adding a point $\backslash infty$ to the complex numbers. ($\backslash mathbf\{CP\}^1$ has the advantage of being compact space.) The basic question is: given a point $z$ in $\backslash mathbf\{CP\}^1$, how does its orbit (or forward orbit)

$z,\backslash ;\; f(z)=z^2,\backslash ;\; f(f(z))=z^4,\; f(f(f(z)))=z^8,\backslash ;\; \backslash ldots$

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| is greater than 1, then the orbit converges to the point $\backslash infty$ in $\backslash mathbf\{CP\}^1$, again more than exponentially fast. (Here 0 and $\backslash infty$ are superattracting fixed points 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.)

On the other hand, suppose that $|z|=1$, 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 set in the circle, and in fact uniformly distributed on the circle. There are also infinitely many on the circle, meaning points with $f^r(z)=z$ for some positive integer r. (Here $f^r(z)$ means the result of applying f to z r times, $f(f(\backslash cdots(f(z))\backslash cdots))$.) 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: if $f^r(z)=z$, the derivative of $f^r$ at z has absolute value greater than 1.)

Pierre Fatou and Gaston Julia showed in the late 1910s that much of this story extends to any complex algebraic map from $\backslash mathbf\{CP\}^1$ to itself of degree greater than 1. (Such a mapping may be given by a polynomial $f(z)$ with complex coefficients, or more generally by a rational function.) Namely, there is always a compact subset of $\backslash mathbf\{CP\}^1$, the Julia set, on which the dynamics of f is chaotic. For the mapping $f(z)=z^2$, 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 is not an integer. This occurs even for mappings as simple as $f(z)=z^2+c$ for a constant $c\backslash in\backslash mathbf\{C\}$. The Mandelbrot set is the set of complex numbers c such that the Julia set of $f(z)=z^2+c$ is connected space.

There is a rather complete classification of the possible dynamics of a rational function $f\backslash colon\backslash mathbf\{CP\}^1\backslash to\; \backslash mathbf\{CP\}^1$ in the Fatou set, the complement of the Julia set, where the dynamics is "tame". Namely, Dennis Sullivan showed that each connected component U of the Fatou set is pre-periodic, meaning that there are natural numbers U, one can assume after replacing f by an iterate that $f(U)=U$. 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.Milnor (2006), section 13. (Note that the "backward orbit" of a point z in U, the set of points in $\backslash mathbf\{CP\}^1$ that map to z under some iterate of f, need not be contained in U.)

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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 $\backslash mathbf\{CP\}^n$ to itself, the richest source of examples. The main results for $\backslash mathbf\{CP\}^n$ have been extended to a class of from any projective variety to itself.Guedj (2010), Theorem B. Note, however, that many varieties have no interesting self-maps.

Let f be an endomorphism of $\backslash mathbf\{CP\}^n$, meaning a morphism of algebraic varieties from $\backslash mathbf\{CP\}^n$ to itself, for a positive integer n. Such a mapping is given in homogeneous coordinates by

$f(z\_0,\backslash ldots,z\_n)=f\_0(z\_0,\backslash ldots,z\_n),\backslash ldots,f\_n(z\_0,\backslash ldots,z\_n)$

for some homogeneous polynomials $f\_0,\backslash ldots,f\_n$ of the same degree d that have no common zeros in $\backslash mathbf\{CP\}^n$. (By Chow's theorem, this is the same thing as a holomorphic mapping from $\backslash mathbf\{CP\}^n$ to itself.) Assume that d is greater than 1; then the degree of the mapping f is $d^n$, which is also greater than 1.

Then there is a unique probability measure $\backslash mu\_f$ on $\backslash mathbf\{CP\}^n$, 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 for $n=1$ (around 1983), and by John Hubbard, Peter Papadopol, John Fornaess, and Nessim Sibony in any dimension (around 1994).Dinh & Sibony (2010), "Dynamics ...", Theorem 1.7.11. The small Julia set $J^*(f)$ is the support of the equilibrium measure in $\backslash mathbf\{CP\}^n$; this is simply the Julia set when $n=1$.

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Examples

• For the mapping $f(z)=z^2$ on $\backslash mathbf\{CP\}^1$, the equilibrium measure $\backslash mu\_f$ is the Haar measure (the standard measure, scaled to have total measure 1) on the unit circle $|z|=1$.
• More generally, for an integer $d>1$, let $f\backslash colon\; \backslash mathbf\{CP\}^n\backslash to\backslash mathbf\{CP\}^n$ be the mapping

:$f(z\_0,\backslash ldots,z\_n)=z\_0^d,\backslash ldots,z\_n^d.$

Then the equilibrium measure $\backslash mu\_f$ is the Haar measure on the n-dimensional torus $\backslash \{1,z\_1,\backslash ldots,z\_n:\; |z\_1|=\backslash cdots=|z\_n|=1\backslash \}.$ For more general holomorphic mappings from $\backslash mathbf\{CP\}^n$ to itself, the equilibrium measure can be much more complicated, as one sees already in complex dimension 1 from pictures of Julia sets.

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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 $f\_*\backslash mu\_f$ is equal to $\backslash mu\_f$. Because f is a finite morphism, the pullback measure $f^*\backslash mu\_f$ is also defined, and $\backslash mu\_f$ is totally invariant in the sense that $f^*\backslash mu\_f=\backslash deg(f)\backslash mu\_f$.

One striking characterization of the equilibrium measure is that it describes the asymptotics of almost every point in $\backslash mathbf\{CP\}^n$ when followed backward in time, by Jean-Yves Briend, Julien Duval, Dinh Tien-Cuong, and Sibony. Namely, for a point z in $\backslash mathbf\{CP\}^n$ and a positive integer r, consider the probability measure $(1/d^\{rn\})(f^r)^*(\backslash delta\_z)$ which is evenly distributed on the $d^\{rn\}$ points w with $f^r(w)=z$. Then there is a Zariski closed subset $E\backslash subsetneq\; \backslash mathbf\{CP\}^n$ such that for all points z not in E, the measures just defined converge weakly to the equilibrium measure $\backslash mu\_f$ as r goes to infinity. In more detail: only finitely many closed complex subspaces of $\backslash mathbf\{CP\}^n$ are totally invariant under f (meaning that $f^\{-1\}(S)=S$), and one can take the exceptional set E to be the unique largest totally invariant closed complex subspace not equal to $\backslash mathbf\{CP\}^n$.Dinh & Sibony (2010), "Dynamics ...", Theorem 1.4.1.

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 $f^r(z)=z$), counted with multiplicity, is $(d^\{r(n+1)\}-1)/(d^r-1)$, which is roughly $d^\{rn\}$. Consider the probability measure which is evenly distributed on the points of period r. Then these measures also converge to the equilibrium measure $\backslash mu\_f$ as r goes to infinity. Moreover, most periodic points are repelling and lie in $J^*(f)$, and so one gets the same limit measure by averaging only over the repelling periodic points in $J^*(f)$.Dinh & Sibony (2010), "Dynamics ...", Theorem 1.4.13. There may also be repelling periodic points outside $J^*(f)$.Fornaess & Sibony (2001), Theorem 4.3.

The equilibrium measure gives zero mass to any closed complex subspace of $\backslash mathbf\{CP\}^n$ that is not the whole space.Dinh & Sibony (2010), "Dynamics ...", Proposition 1.2.3. Since the periodic points in $J^*(f)$ are dense in $J^*(f)$, it follows that the periodic points of f are Zariski dense in $\backslash mathbf\{CP\}^n$. A more algebraic proof of this Zariski density was given by Najmuddin Fakhruddin.Fakhruddin (2003), Corollary 5.3. Another consequence of $\backslash mu\_f$ giving zero mass to closed complex subspaces not equal to $\backslash mathbf\{CP\}^n$ is that each point has zero mass. As a result, the support $J^*(f)$ of $\backslash mu\_f$ has no isolated points, and so it is a perfect set.

The support $J^*(f)$ of the equilibrium measure is not too small, in the sense that its Hausdorff dimension is always greater than zero. 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 $J^*(f)$ is all of $\backslash mathbf\{CP\}^n$.Milnor (2006), Theorem 5.2 and problem 14-2; Fornaess (1996), Chapter 3.) 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 $n\backslash log\; d$, by Mikhail Gromov, Michał Misiurewicz, and Feliks Przytycki.Dinh & Sibony (2010), "Dynamics ...", Theorem 1.7.1.

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 $\backslash mathbf\{CP\}^n$, the equilibrium measure $\backslash mu\_f$ 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.Dinh & Sibony (2010), "Dynamics ...", Theorem 1.6.3. It follows, for example, that for almost every point with respect to $\backslash mu\_f$, its forward orbit is uniformly distributed with respect to $\backslash mu\_f$.

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Lattès maps A Lattès map is an endomorphism f of $\backslash mathbf\{CP\}^n$ 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 on $\backslash mathbf\{CP\}^n$. Conversely, by Anna Zdunik, François Berteloot, and Christophe Dupont, the only endomorphisms of $\backslash mathbf\{CP\}^n$ whose equilibrium measure is absolutely continuous with respect to Lebesgue measure are the Lattès examples.Berteloot & Dupont (2005), Théorème 1. That is, for all non-Lattès endomorphisms, $\backslash mu\_f$ 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 of a probability measure $\backslash mu$ on $\backslash mathbf\{CP\}^1$ (or more generally on a smooth manifold) by

$\backslash dim(\backslash mu)=\backslash inf\; \backslash \{\backslash dim\_H(Y):\backslash mu(Y)=1\backslash \},$

where $\backslash dim\_H(Y)$ denotes the Hausdorff dimension of a Borel set Y. For an endomorphism f of $\backslash mathbf\{CP\}^1$ of degree greater than 1, Zdunik showed that the dimension of $\backslash mu\_f$ 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 (up to sign), or a power map $f(z)=z^\{\backslash pm\; d\}$ with $d\backslash geq\; 2$.Zdunik (1990), Theorem 2; Berteloot & Dupont (2005), introduction. (In the latter cases, the Julia set is all of $\backslash mathbf\{CP\}^1$, a closed interval, or a circle, respectively.Milnor (2006), problem 5-3.) 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.

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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 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 $H^*(X,\backslash mathbf\{Z\})$.

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.Cantat (2000), Théorème 2.2. Explicitly, for X of complex dimension n and $0\backslash leq\; p\backslash leq\; n$, let $d\_p$ be the spectral radius of f acting by pullback on the Hodge theory group $H^\{p,p\}(X)\backslash subset\; H^\{2p\}(X,\backslash mathbf\{C\})$. Then the topological entropy of f is

$h(f)=\backslash max\_p\; \backslash log\; d\_p.$

(The topological entropy of f is also the logarithm of the spectral radius of f on the whole cohomology $H^*(X,\backslash mathbf\{C\})$.) 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 and K3 surfaces do have such automorphisms.Cantat (2010), sections 7 to 9.

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 $d\_p$ takes its maximum value, the action of f on $H^\{p,p\}(X)$ has only one eigenvalue with absolute value $d\_p$, 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.Cantat (2014), section 2.4.3. (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 $\backslash mu\_f$ of maximal entropy for f, called the equilibrium measure (or Green measure, or measure of maximal entropy).De Thélin & Dinh (2012), Theorem 1.2. (In particular, $\backslash mu\_f$ has entropy $\backslash log\; d\_p$ with respect to f.) The support of $\backslash mu\_f$ is called the small Julia set $J^*(f)$. Informally: f has some chaotic behavior, and the most chaotic behavior is concentrated on the small Julia set. At least when X is projective, $J^*(f)$ has positive Hausdorff dimension. (More precisely, $\backslash mu\_f$ assigns zero mass to all sets of sufficiently small Hausdorff dimension.)Dinh & Sibony (2010), "Super-potentials ...", section 4.4.

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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 $E\backslash times\; E$. Then the group $GL(2,\backslash mathbf\{Z\})$ of invertible $2\backslash times\; 2$ integer matrices acts on X. Any group element f whose trace has absolute value greater than 2, for example , 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.Cantat & Dupont (2020), section 1.2.1.

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 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.Cantat & Dupont (2020), Main Theorem. In this sense, it is usual for the equilibrium measure of an automorphism to be somewhat irregular.

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Saddle periodic points A periodic point z of f is called a saddle periodic point if, for a positive integer r such that $f^r(z)=z$, at least one eigenvalue of the derivative of $f^r$ 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 with simple action on cohomology, the saddle periodic points are dense in the support $J^*(f)$ of the equilibrium measure $\backslash mu\_f$. On the other hand, the measure $\backslash mu\_f$ vanishes on closed complex subspaces not equal to X. It follows that the periodic points of f (or even just the saddle periodic points contained in the support of $\backslash mu\_f$) 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 $\backslash mu\_f$.Dinh & Sibony (2010), "Super-potentials ...", Theorem 4.4.2. It follows that for almost every point z with respect to $\backslash mu\_f$, the forward and backward orbits of z are both uniformly distributed with respect to $\backslash mu\_f$.

A notable difference with the case of endomorphisms of $\backslash mathbf\{CP\}^n$ 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 $J^*(f)$ 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.Cantat (2010), Théorème 9.8. Points in that open set never approach $J^*(f)$ 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 $h(f)=\backslash log\; d\_1$. Consider the probability measure which is evenly distributed on the isolated periodic points of period r (meaning that $f^r(z)=z$). Then this measure converges weakly to $\backslash mu\_f$ as r goes to infinity, by Eric Bedford, Lyubich, and John Smillie.Cantat (2014), Theorem 8.2. The same holds for the subset of saddle periodic points, because both sets of periodic points grow at a rate of $(d\_1)^r$.

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See also

• Dynamics in complex dimension 1
• Complex analysis
• Complex quadratic polynomial
• Infinite compositions of analytic functions
• Montel's theorem
• Poincaré metric
• Schwarz lemma
• Riemann mapping theorem
• Carathéodory's theorem (conformal mapping)
• Böttcher's equation
• Yoccoz puzzles

• Related areas of dynamics
• Arithmetic dynamics
• Chaos theory
• Symbolic dynamics

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Notes

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External links

• Gallery of dynamics (Curtis McMullen)
• Surveys in Dynamical Systems

Categories: Complex Analysis, Dynamical Systems, Chaos Theory, Fractals

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