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...