# Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

Araceli Bonifant
Mikhail Lyubich
Scott Sutherland
Pages: 824
https://www.jstor.org/stable/j.ctt5vjv7b

1. Front Matter
(pp. i-iv)
(pp. v-x)
3. Preface
(pp. xi-xii)
4. Introduction
(pp. 1-12)

Holomorphic dynamicsis one of the earliest branches of dynamical systems which is not part of classical mechanics. As a prominent field in its own right, it was founded in the classical work of Fatou and Julia (see [Fa1, Fa2] and [J]) early in the 20th century. For some mysterious reason, it was then almost completely forgotten for 60 years. The situation changed radically in the early 1980s when the field was revived and became one of the most active and exciting branches of mathematics. John Milnor was a key figure in this revival, and his fascination with holomorphic dynamics...

5. ### Part I. One Complex Variable

• Arithmetic of Unicritical Polynomial Maps
(pp. 15-24)
John Milnor

A complex polynomial maps of degree$n\ge2$with only one critical point can always be put in the standard normal form

${f_c}(z)={z^n}+c$(1.1)

by an affine change of coordinates. The connectedness locus, consisting of allcfor which the Julia set of${f_c}$is connected, is sometimes known as the “multibrot set.” (Compare [S].) It is not difficult to check that the power$\hat c={c^{n-1}}$is a complete invariant for the holomorphic conjugacy class of${f_c}$

In §2 we will use the alternate normal form

$w\mapsto{g_b}(w)=\frac{{{w^n}+b}}{n}$, (1.2)

with derative${g'_b}(w)={w^{n-1}}$, and use the conjugacy invariant$\hat b={b^{n-1}}$. These normal forms (1.1) and...

• Les racines des composantes hyperboliques de M sont des quarts d’entiers algébriques
(pp. 25-26)
Thierry Bousch

Théoréme.Soit c un nombre complexe, et z un point périodique du polynôme$z \mapsto {z^2} + c$.Le multiplicateur ρ de l’orbite de z est un entier algébrique si et seulement si4c est un entier algébrique.

Corollaire.Si c est une racine de composante hyperbolique de l’ensemble de Mandelbrot, alors4c est un entier algébrique.

Notations. On pose$C = 4c,{\text{et }}f(z) = {z^2} + c$. Considérons l’équation${f^n}(z) = z$comme équation en z sur le corps$K = \mathbb{C}(c)$; cette équation admet$N = {2^n}$solutionsdistinctesdans la clôture algébrique deK, que nous noterons${z_1},...,{z_N}$. (Ceci correspond au fait que pourcnombre complexe générique, l’application quadratique$z \mapsto {z^2} + c$posséde exactementN...

• Dynamical cores of topological polynomials
(pp. 27-48)
Alexander Blokh, Lex Oversteegen, Ross Ptacek and Vladlen Timorin

Complex dynamics studies, among other topics, the limiting behavior of points under iteration of complex polynomials. It is reasonable to restrict our attention to the Julia set, as elsewhere the limiting behavior is easy to describe. Since in many cases polynomial Julia sets are one-dimensional continua, one can consider the problem as a far-reaching generalization of the similar problem for simple one-dimensional spaces such as an interval.

A popular one-dimensional family is that ofunimodal interval maps, i.e., interval maps with unique turning point. Often such mapsfare considered on [0, 1] and normalized by assuming that the turning...

• The quadratic dynatomic curves are smooth and irreducible
(pp. 49-72)
Xavier Buff and Tan Lei

For$c \in \mathbb{C}$, let${f_c}:\mathbb{C} \to \mathbb{C}$be the quadratic polynomial

${f_c}(z): = {z^2} + c$.

A point$z \in \mathbb{C}$is periodic for${f_c}$if$f_c^{on}(z) = z$for some integer$n\underline { > 1}$; it is of periodnif$f_c^{ok}(z) \ne z$for$0 < k < n$. For$n\underline > 1$, let${X_n} \subset {\mathbb{C}^2}$, be thedynatomic curve defined by

${X_n}: = \{ (c,z) \in \left. {{\mathbb{C}^2}} \right|z$is periodic of periodnfor${f_c}\}$.

The objective of this note is to give new proofs of the following known results.

Theorem 1.1 (Douady-Hubbard).For every$n\underline > 1$,the closure of${X_n}$in${\mathbb{C}^2}$is a smooth affine curve.

Theorem 1.2 (Bousch and Lau-Schleicher).For every$n\underline > 1$the closure of${X_n}$in${\mathbb{C}^2}$is irreducible.

Theorem 1.1 has...

• Multicorns are not path connected
(pp. 73-102)
John Hamal Hubbard and Dierk Schleicher

Themulticorn${\cal M}_d^*$is the connectedness locus in the space of antiholomorphic unicritical polynomials${p_c}(z) = {z^{ - d}} + c$of degreed, i.e., the set of parameters for which the Julia set is connected. The special case$d = 2$is thetricorn,which is the formal antiholomorphic analog to the Mandelbrot set.

The second iterate is

$p_c^{ \circ 2}(z) = {\overline {({z^{ - d}} + c)} ^d} + c = {({z^d} + \bar c)^d} + c$

and thus is holomorphic in the dynamical variablezbut no longer complex analytic in the parameterc.Much of the dynamical theory of antiholomorphic polynomials (in short, antipolynomials) is thus in analogy to the theory of holomorphic polynomials, except for certain features near periodic points of odd...

• Leading monomials of escape regions
(pp. 103-120)
Jan Kiwi

This note will use non-Archimedean methods to prove a result in classical holomorphic dynamics. In particular, we will prove a result in complex cubic polynomial dynamics using cubic polynomial dynamics over the field of formal Puiseux series.

The space of complex cubic polynomials, regarded as dynamical systems acting on the complex plane$\mathbb{C}$, has two complex dimensions. To understand how polynomials are organized in parameter space according to dynamics, it is of interest to study complex one-dimensional slices. Our focus in this paper is on slices formed by cubic polynomials with a periodic critical point. These slices have already received...

• Limiting behavior of Julia sets of singularly perturbed rational maps
(pp. 121-134)
Robert L. Devaney

In recent years there have been a number of papers dealing with singular perturbations of complex dynamical systems. Most of these papers deal with maps of the form${z^n} + c + \lambda /{z^d}$, where$n\underline > 2$and$d\underline > 1$andcis the center of a hyperbolic component of the multibrot set, i.e., the connectedness locus for the family${z^n} + c$. These maps are called singular perturbations because, when$\lambda = 0$, the map is just the polynomial${z^n} + c$and the dynamical behavior for this map is completely understood. When$\lambda \ne 0$, the degree of the map changes and the dynamical behavior suddenly explodes.

Our goal in this paper is...

• On (non-)local connectivity of some Julia sets
(pp. 135-162)
Alexandre Dezotti and Pascale Roesch

In this note we discuss the following question: When is the Julia set of a rational map connected but not locally connected? We propose some conjectures and develop a model of non-locally connected Julia sets in the case of infinitely renormalizable quadratic polynomials, a situation where one hopes to find a precise answer.

The question of local connectivity of the Julia set has been studied extensively for quadratic polynomials, but there is still no complete characterization of when a quadratic polynomial has a connected and locally connected Julia set. In degree 2, the question reduces to the precise cases where...

• Perturbations of weakly expanding critical orbits
(pp. 163-196)

We say that a critical pointcof a rational functionfisweakly expanding, or summable, if, for the point$\upsilon = f(c)$of the Riemann sphere$\bar \mathbb{C}$,

$\sum\limits_{n = 0}^\infty {\frac{{1 + {{\left| {{f^n}(\upsilon )} \right|}^2}}} {{1 + {{\left| \upsilon \right|}^2}}}} \frac{1} {{\left| {({f^n})'(\upsilon )} \right|}} < \infty$

Throughout the paper, derivatives are standard derivatives of holomorphic maps; then the summand in (1.1) is a finite number for every$\upsilon \in \overline \mathbb{C}$as soon as$\upsilon$is not a critical point of${f^n}$.

In the present paper, we study perturbations of polynomials and rational functions with several (possibly, not all) summable critical points. This paper is a natural continuation of [Le02], [Le11], and, partly, [Le88] and [LSY]. Let us the state main...

• Unmating of rational maps: Sufficient criteria and examples
(pp. 197-234)
Daniel Meyer

Douady and Hubbard observed that often “one can find” Julia sets of polynomials within Julia sets of rational maps. This prompted them to introduce the operation ofmating of polynomials; see [Dou83]. It is a way to glue together two (connected and locally connected) filled Julia sets along their boundaries. The dynamics then descends to the quotient. Somewhat surprisingly, one often obtains a map that is topologically conjugate to a rational map.

Thurston’s celebrated theorem on the classification of rational maps among (postcritically finite) topological rational maps (see [DH93]) can be applied to answer the question of when the mating...

• A framework toward understanding the characterization of holomorphic dynamics
(pp. 235-258)
Yunping Jiang

Suppose$f:{S^2} \to {S^2}$is an orientation-preserving branched covering of the two-sphere${S^2}$, often simply referred to as a “branched covering.” We use the notation${f^n} = \underbrace {fo \cdots of}_n$for the n-fold composition. Letd= degf> 1 be the degree of and let

${\Omega _f} = \{ c\left| {{{\deg }_c}f\underline > 2} \right.\}$

be the set of all branched points, where${\deg _c}f$means the local degree offatc. Let

${P_f} = \overline {{U_{n\underline > 1}}{f^n}({\Omega _f})}$

be the post-critical set. A rational map which is a quotient of two polynomials is a holomorphic branched covering.

Definition 1.1 (Combinatorial equivalence).Two branched coverings f and g are said to be combinatorially equivalent if there are two...

6. ### Part II. One Real Variable

• Metric stability for random walks (with applications in renormalization theory)
(pp. 261-322)
Carlos Gustavo Moreira and Daniel Smania

In the study of a dynamical system, some of the most important questions concerns the stability of their dynamical properties under (most of the) perturbations: how robust are they?

Here we are mainly interested in the stability of metric (measure-theoretic) properties of dynamical systems. A well-known example is that of${C^2}$expanding maps on the circle; this class is structurally stable, and all such maps have an absolutely continuous and ergodic invariant probability satisfying certain decay of correlations estimates. In particular, in the measure theoretic sense, most of the orbits are dense in the phase space.

Now let us study...

• Milnor’s conjecture on monotonicity of topological entropy: Results and questions
(pp. 323-338)
Sebastian van Strien

In their seminal and widely circulated 1977 preprint, “On iterated maps of the interval: I,II,” Milnor and Thurston proved the following.

Theorem (Milnor and Thurston [MT77], see also [MT88]).The function${C^{2,b}} \to \mathbb{R}$which associates to a mapping$g \in {C^{2,b}}$its topological entropy${h_{top}}(g)$is continuous.

Here${C^{2,b}}$stands for${C^2}$maps of the interval with b non-degenerate critical points (non-degenerate means second derivative non-zero). This theorem relies on a result of Misiurewicz and Szlenk [MS77, MS80], who had previously shown that$\gamma (f): = \exp ({h_{top}}(f))$is equal to the growth rate of the number of laps (i.e., intervals of monotonicity) of${f^n}$. The crucial...

• Entropy in dimension one
(pp. 339-384)
William P. Thurston

The topological entropy$h(f)$of a map from a compact topological space to itself,$f:X \to X$, is a numerical measure of the unpredictability of trajectories$x,f(x),{f^2}(x)$, . . . of points underf: it is the limiting upper bound for exponential growth rate of$\varepsilon$-distinguishable orbits, as$\varepsilon \to 0$. Here$\varepsilon$can be measured with respect to an arbitrary metric onX, or one can merely think of it as a neighborhood of the diagonal in$X \times X$, since all that matters is whether or not two points are within$\varepsilon$of each other. The number of$\varepsilon-distinguishable orbits of lengthnis... 7. ### Part III. Several Complex Variables • On Écalle-Hakim’s theorems in holomorphic dynamics (pp. 387-450) Marco Arizzi and Jasmin Raissy One of the main questions in the study of local discrete holomorphic dynamics, i.e., in the study of the iterates of a germ of a holomorphic map of${\mathbb{C}^p}$at a fixed point, which can be assumed to be the origin, is when it is possible to holomorphically conjugate it to a “simple” form, possibly its linear term. It turns out (see [Ab3], [Ab4], [Br], [CC], [IY], [Yo] and Chapter 1 of [Ra] for general surveys on this topic) that the answer to this question strongly depends on the arithmetical properties of the eigenvalues of the linear term of the... • Index theorems for meromorphic self-maps of the projective space (pp. 451-462) Marco Abate In this short note we would like to show how the techniques introduced in [ABT1] (see also [ABT2], [ABT3], [Br], [BT] and [AT2]) for studying the local dynamics of holomorphic germs tangent to the identity can be used to study global meromorphic self-maps of the complex projective space${\mathbb{P}^n}$. More precisely, we shall prove the following index theorem. Theorem 1.1. Let$f:{\mathbb{P}^n} \to {\mathbb{P}^n}$be a meromorphic self-map of degree$v + 1 \ge 2$of the complex n-dimensional projective space. Let$\sum (f) = Fix(f) \cup I(f)$be the union of the indeterminacy set I(f) of f and the fixed points set Fix(f) of f. Let$\sum {(f)} = { \sqcup _\alpha }{\sum _\alpha }$be the decomposition... • Dynamics of automorphisms of compact complex surfaces (pp. 463-514) Serge Cantat LetMbe a compact complex manifold. By definition, holomorphic diffeomorphisms$f:M \to M$are called automorphisms; they form a group, the group Aut(M) of automorphisms ofM. Endowed with the topology of uniform convergence, Aut(M) is a topological group, and a theorem due to Bochner and Montgomery shows that this topological group is a complex Lie group, whose Lie algebra is the algebra of holomorphic vector fields onM(see [18]). The connected component of the identity in Aut(M) is denoted Aut(M)⁰, and the group of its connected components is Aut(M)#= Aut(M)/Aut(M)⁰. If$M = {\mathbb{P}^1}(\mathbb{C})$, then Aut(M) is$PG{L_2}(\mathbb{C})$, the group... • Bifurcation currents and equidistribution in parameter space (pp. 515-566) Romain Dujardin Let${({f_\lambda })_{\lambda \in \Lambda }}$be a holomorphic family of dynamical systems acting on the Riemann sphere${\mathbb{P}^1}$, parameterized by a complex manifold$\Lambda $. The “dynamical systems” in consideration here can be polynomial or rational mappings on${\mathbb{P}^1}$, as well as groups of Möbius transformations. It is a very basic idea that the product dynamics$\hat f$acting on$\Lambda \times {\mathbb{P}^1}$by$\hat f(\lambda ,z) = (\lambda ,{f_\lambda }(z))$is an important source of information on the bifurcation theory of the family. The input of techniques from higherdimensional holomorphic dynamics dynamics into this problem recently led to a number of interesting new results in this area, especially when the parameter space... 8. ### Part IV. Laminations and Foliations • Entropy for hyperbolic Riemann surface laminations I (pp. 569-592) Tien-Cuong Dinh, Viet-Anh Nguyên and Nessim Sibony The main goal of this paper is to introduce a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. We also study the transverse regularity of the Poincaré metric and the finiteness of the entropy. In order to simplify the presentation, we will mostly focus, in this first part, on compact laminations which are transversally smooth. We will study the case of singular foliations in the second part of this paper. The question of hyperbolicity of leaves for generic foliations in${\mathbb{P}^k}$has been adressed by many authors. We just mention here the case of a polynomial vector... • Entropy for hyperbolic Riemann surface laminations II (pp. 593-622) Tien-Cuong Dinh, Viet-Anh Nguyên and Nessim Sibony In this second part, we study Riemann surface foliations with tame singular points. We say that a holomorphic vector fieldFin${\mathbb{C}^k}$isgeneric linearif it can be written as$F(z) = \sum\limits_{j = 1}^k {{\lambda _j}{z_j}} \frac{\partial }{{\partial {z_j}}}$where the${\lambda _j}$are non-zero complex numbers. The integral curves ofFdefine a Riemann surface foliation in${\mathbb{C}^k}$. The condition${\lambda _j} \ne 0$for everyjimplies that the foliation has an isolated singularity at 0. Consider a Riemann surface foliation with singularities$(X,\mathcal{L},E)\$in a complex manifoldX. We assume that the singular setEis discrete. Byfoliation, we mean thatLis transversally holomorphic. We...

• Intersection theory for ergodic solenoids
(pp. 623-644)
Vicente Muñoz and Ricardo Pérez-Marco

In [3], the authors define the concept ofk-solenoid as an abstract laminated space and prove that a solenoid with a transversal measure immersed in a smooth manifold defines a generalized Ruelle-Sullivan current, as considered in [8]. Considering as object the abstract space with the inmersion provides a more versatile concept than previously considered by other authors (see, for example, [1], [9], or [10]) that is suitable for results like the representation theorem in [5]. The purpose of the current paper is to study the intersection theory of such objects.

IfMis a smooth manifold, any closed oriented submanifold...

• Invariants of four-manifolds with flows via cohomological field theory
(pp. 645-676)
Hugo García-Compeán, Roberto Santos-Silva and Alberto Verjovsky

Quantum field theory is not only a framework to describe the physics of elementary particles and condensed matter systems, but it has been useful to describe mathematical structures and their subtle interrelations. One of the most famous examples is perhaps the description of knot and link invariants through the correlation functions of products of Wilson line operators in the Chern-Simons gauge theory [1]. These invariants are the Jones-Witten invariants or Vassiliev invariants, depending on whether the coupling constant is weak or strong, respectively. Very recently some aspects of gauge and string theories found a strong relation with Khovanov homology [2]....

• Color Plates
(pp. None)
9. ### Part V. Geometry and Algebra

• Two papers which changed my life: Milnor’s seminal work on flat manifolds and bundles
(pp. 679-704)
William M. Goldman

For a young student studying topology at Princeton in the mid-1970s, John Milnor was a inspiring presence. The excitement of hearing him lecture at the Institute for Advanced Study and reading his books and unpublished lecture notes available in Fine Library made a deep impact on me. One heard rumors of exciting breakthroughs in the Milnor-Thurston collaborations on invariants of 3-manifolds and the theory of kneading in 1-dimensional dynamics. The topological significance of volume in hyperbolic 3-space and Gromov’s proof of Mostow rigidity using simplicial volume were in the air at the time (later to be written up in Thurston’s...

• Milnor’s problem on the growth of groups and its consequences
(pp. 705-774)
Rostislav Grigorchuk

The notion of the growth of a finitely generated group was introduced by A. S. Schwarz (also spelled Schvarts and Švarc) [Š55] and independently by Milnor [Mil68b, Mil68a]. Particular studies of group growth and their use in various situations have appeared in the works of Krause [Kra53], Adelson-Velskii and Shreider [AVŠ57], Dixmier [Dix60], Dye [Dye59], [Dye63], Arnold and Krylov [AK63], Kirillov [Kir67], Avez [Ave70], Guivarc’h [Gui70], [Gui71], [Gui73], Hartley, Margulis, Tempelman, and other researchers. The note of Schwarz did not attract a lot of attention in the mathematical community and was essentially unknown to mathematicians both in the USSR and...

10. Contributors
(pp. 775-778)
11. Index
(pp. 779-787)