Holomorphic dynamics is 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 helped to make it so prominent. Milnor's book Dynamics in One Complex Variable [M8], his volumes of collected papers [M10, M11], and the surveys [L1, L5] are exemplary introductions into the richness and variety of Milnor's work in dynamics.

Holomorphic dynamics, in the sense we will use the term here, studies iterates of holomorphic maps on complex manifolds. Classically, it focused on the dynamics of rational maps of the Riemann sphere [??]. For such a map f, the Riemann sphere is decomposed into two invariant subsets, the Fatou set F(f), where the dynamics is quite tame, and the Julia set J(f), which often has a quite complicated fractal structure and supports chaotic dynamics.

Even in the case of quadratic polynomials Qc : z [??] z2 + c, the dynamical picture is extremely intricate and may depend on the parameter c in an explosive way. The corresponding bifurcation diagram in the parameter plane is called the Mandelbrot set; its first computer images appeared in the late 1970s, sparking an intense interest in the field [BrMa, Man].

The field of holomorphic dynamics is rich in interactions with many branches of mathematics, such as complex analysis, geometry, topology, number theory, algebraic geometry, combinatorics, and measure theory. The present book is a clear example of such interplay.

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The papers "Arithmetic of Unicritical Polynomial Maps" and "Les racines de composantes hyperboliques de M sont des quarts d'entiers algébriques," which open this volume, exemplify the interaction of holomorphic dynamics with number theory. In these papers, John Milnor and Thierry Bousch study number-theoretic properties of the family of polynomials pc(z) = zn + c, whose bifurcation diagram is known as the Multibrot set.

In the celebrated Orsay Notes [DH1], Douady and Hubbard undertook a remarkable combinatorial investigation of the Mandelbrot set and the corresponding bifurcations of the Julia sets. In particular, they realized (using important contributions from Thurston's work [T]) that these fractal sets admit an explicit topological model as long as they are locally connected (see [D]). This led to the most famous conjecture in the field, on the local connectivity of the Mandelbrot set, typically abbreviated as MLC. The MLC conjecture is still currently open, but it has led to many important advances, some of which are reflected in this volume.

In his thesis [La], Lavaurs proved the non-local-connectivity of the cubic connectedness locus, highlighting the fact that the degree two case is special in this respect. In attempt to better understand this phenomenon, Milnor came across a curious new object that he called the tricorn: the connectedness locus of antiholomorphic quadratic maps qc(z) = [bar.z]2 + c. In the paper "Multicorns are not path connected," John Hubbard and Dierk Schleicher take a close look at the connectedness locus of its higher degree generalization, defined by pc(z) = [bar.z]n + c.

The paper by Alexandre Dezotti and Pascale Roesch, "On (non-)local connectivity of some Julia sets," surveys the problem of local connectivity of Julia sets. It collects a variety of results and conjectures on the subject, both "positive" and "negative" (as Julia sets sometimes fail to be locally connected). In particular, in this paper the reader can learn about the work of Yoccoz [H, M7], Kahn and Lyubich [KL], and Kozlovski, Shen, and van Strien [KSvS]; the latter gives a positive answer in the case of "non-renormalizable" polynomials of any degree.

Related to connectivity, an important question that has interested both complex and algebraic dynamicists is that of the irreducibility of the closure of Xn, the set of points (c, z) [member of] C2 for which z is periodic under Qc(z) = z2 + c with minimal period n. These curves are known as dynatomic curves. The irreducibility of such curves was proved by Morton [Mo] using algebraic methods, by Bousch [Bou] using algebraic and analytic (dynamical) methods, and by Lau and Schleicher [LS], using only dynamical methods. In the paper "The quadratic dynatomic curves are smooth and irreducible," Xavier Buff and Tan Lei present a new proof of this result based on the transversality theory developed by Adam Epstein [E].

Similarly, in the case of the family of cubic polynomial maps with one marked critical point, parametrized by the equation F(z) = z3 - 3a2z + (2a3 + v), one can study the period p-curves Sp for p ≥ 1. These curves are the collection of parameter pairs (a, v) [member of] C2 for which the marked critical point a has period exactly p; Milnor proved that Sp is smooth and affine for all p > 0 and irreducible for p ≤ 3 [M9]. The computation of the Euler characteristic for any p > 0 and the irreducibility for p = 4 were proved by Bonifant, Kiwi and Milnor [BKM]. The computation of the Euler characteristic requires a deep study of the unbounded hyperbolic components of Sp, known as escape regions. Important information about the limiting behavior of the periodic critical orbit as the parameter tends to infinity within an escape region is encoded in an associated leading monomial vector, which uniquely determines the escape region, as Jan Kiwi shows in "Leading monomials of escape regions."

As we have alluded to previously, a locally connected Julia set admits a precise topological model, due to Thurston, by means of a geodesic lamination in the unit disk. This model can be efficiently described in terms of the Hubbard tree, which is the "core" that encodes the rest of the dynamics. In particular, it captures all the cut-points of the Julia set, which generate the lamination in question. This circle of ideas is described and is carried further to a more general topological setting in the paper by Alexander Blokh, Lex Oversteegen, Ross Ptacek and Vladlen Timorin "Dynamical cores of topological polynomials."

The realm of general rational dynamics on the Riemann sphere is much less explored than that of polynomial dynamics. There is, however, a beautiful bridge connecting these two fields called mating: a surgery introduced by Douady and Hubbard in the 1980s, in which the filled Julia sets of two polynomials of the same degree are dynamically related via external rays. In many cases this process produces a rational map. It is a difficult problem to decide when this surgery works and which rational maps can be obtained in this way. A recent breakthrough in this direction was achieved by Daniel Meyer, who proved that in the case when f is postcritically finite and the Julia set of f is the whole Riemann sphere, every sufficiently high iterate of the map can be realized as a mating [Me1, Me2]. In the paper "Unmating of rational maps, sufficient criteria and examples," Meyer gives an overview of the current state of the art in this area of research, illustrating it with many examples. He also gives a sufficient condition for realizing rational maps as the mating of two polynomials.

Another way of producing rational maps is by "singular" perturbations of complex polynomials. In the paper "Limiting behavior of Julia sets of singularly perturbed rational maps," Robert Devaney surveys dynamical properties of the families fc,λ(z) = zn + c + λ/zd for n ≥ 2, d ≥ 1, with c corresponding to the center of a hyperbolic component of the Multibrot set. These rational maps produce a variety of interesting Julia sets, including Sierpinski carpets and Sierpinski gaskets, as well as laminations by Jordan curves. In the current article, the author describes a curious "implosion" of the Julia sets as a polynomial pc = zn + c is perturbed to a rational map fc,λ.

There is a remarkable phase-parameter relation between the dynamical and parameter planes in holomorphic families of rational maps. It first appeared in the early 1980s in the context of quadratic dynamics in the Orsay notes [DH1] and has become a very fruitful philosophy ever since. In the paper "Perturbations of weakly expanding critical orbits," Genadi Levin establishes a precise form of this relation for rational maps with one critical point satisfying the summability condition (certain expansion rate assumption along the critical orbit). This result brings to a natural general form many previously known special cases studied over the years by many people, including the author.

One of the most profound achievements in holomorphic dynamics in the early 1980s was Thurston's topological characterization of rational maps, which gives a combinatorial criterion for a postcritically finite branched covering of the sphere to be realizable (in a certain homotopical sense) as a rational map (see [DH2]). A wealth of new powerful ideas from hyperbolic geometry and Teichmüller theory were introduced to the field in this work. The Thurston Rigidity Theorem, which gives uniqueness of the realization, although only a small part of the theory, already is a major insight, with many important consequences for the field (some of which are mentioned later).

Attempts to generalize Thurston's characterization to the transcendental case faces many difficulties. However, in the exponential family z [??] eλz, they were overcome by Hubbard, Schleicher and Shishikura [HSS]. In the paper "A framework towards understanding the characterization of holomorphic dynamics," Yunping Jiang surveys these and further results, which, in particular, extend the theory to a certain class of postcritically infinite maps. His paper includes an appendix by the author, Tao Chen, and Linda Keen that proposes applications of the ideas developed on the survey to the characterization problem for certain families of quasi-entire and quasi-meromorphic functions.

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The field of real one-dimensional dynamics emerged from obscurity in the mid-1970s, largely due to the seminal work by Milnor and Thurston [MT], where they laid down foundations of the combinatorial theory of one-dimensional dynamics, called kneading theory. To any piecewise monotone interval map f, the authors associated a topological invariant (determined by the ordering of the critical orbits on the line) called the kneading invariant, which essentially classifies the maps in question. Another important invariant, the topological entropy h(f) (which measures "the complexity" of a dynamical system) can be read off from the kneading invariant. One of the conjectures posed in the preprint version of [MT] was that in the real quadratic family fa : x [??] ax (1 - x), a [member of] (0, 4], the topological entropy depends monotonically on a. This conjecture was proved in the final version using methods of holomorphic dynamics (the Thurston Rigidity Theorem alluded to earlier). This was the first occasion that demonstrated how fruitful complex methods could be in real dynamics. Much more was to come: see, e.g., [L4], a recent survey on this subject.

Later on, Milnor posed the general monotonicity conjecture [M6] (compare [DGMTr]) asserting that in the family of real polynomials of any degree, isentropes are connected (where an isentrope is the set of parameters with the same entropy). This conjecture was proved in the cubic case by Milnor and Tresser [MTr], and in the general case by Bruin and van Strien [BvS]. In the survey "Milnor's conjecture on monotonicity of topological entropy: Results and questions," Sebastian van Strien discusses the history of this conjecture, gives an outline of the proof in the general case, and describes the state of the art in the subject. The proof makes use of an important result by Kozlovski, Shen, and van Strien [KSvS] on the density of hyperbolicity in the space of real polynomial maps, which is a far-reaching generalization of the Thurston Rigidity Theorem. (In the quadratic case, density of hyperbolicity had been proved in [L3, GrSw].) The article concludes with a list of open problems.

The paper "Entropy in dimension one" is one of the last papers written by William Thurston and occupies a special place in this volume. Sadly, Bill Thurston passed away in 2012 before finishing this work. In this paper, Thurston studies the topological entropy h of postcritically finite one-dimensional maps and, in particular, the relations between dynamics and arithmetics of eh, presenting some amazing constructions for maps with given entropy and characterizing what values of entropy can occur for postcritically finite maps. In particular, he proves: h is the topological entropy of a postcritically finite interval map if and only if h = log λ, where λ ≥ 1 is a weak Perron number, i.e., it is an algebraic integer, and λ ≥ |λσ| for every Galois conjugate λσ [member of] C.

The editors received significant help from many people in preparing this paper for publication, to whom we are very grateful. Among them are M. Bestvina, M. Handel, W. Jung, S. Koch, D. Lind, C. McMullen, L. Mosher, and Tan Lei. We are especially grateful to John Milnor, who carefully studied Bill's manuscript, adding a number of notes which clarify many of the points mentioned in the paper.

In the mid-1980s, Milnor wrote a short conceptual article "On the concept of attractor" [M5] that made a substantial impact on the field of real one-dimensional dynamics. In this paper Milnor proposed a general notion of measure-theoretic attractor, illustrated it with the Feigenbaum attractor, and formulated a problem of existence of wild attractors in dimension one. Such an attractor would be a Cantor set that attracts almost all orbits of some topologically transitive periodic interval. It turns out that the answer depends on the degree: in the quadratic case, there are no wild attractors [L2], while they can exist for higher degree unicritical maps x [??] xd + c [BKNvS]. This work made use of the idea of random walk, which describes transitions between various dynamical scales. In the paper "Metric stability for random walks (with applications in renormalization theory)," by Carlos Moreira and Daniel Smania, this idea was carried further to prove a surprising rigidity result: the conjugacy between two unimodal maps of the same degree with Feigenbaum or wild attractors is absolutely continuous.

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One of the central results of classical local one-dimensional holomorphic dynamics is the Leau-Fatou Flower Theorem describing the local dynamics near a parabolic point (see [M8]). A natural problem is to develop a similar theory in higher dimensions. In the late 1990s important results in this direction were obtained by M. Hakim [Ha1, Ha2], but unfortunately, they are only partially published. In the paper "On Écalle-Hakim's theorems in holomorphic dynamics," Marco Arizzi and Jasmin Raissy give a detailed technical account of these advances. This paper is followed by a note, "Index theorems for meromorphic self-maps of the projective space," by Marco Abate, in which local techniques are used to prove three index theorems for global meromorphic maps of projective space.