This will be a crash course in the theory of iteration of rational maps. It is oriented to undergraduates and starting graduate students attending our seminar.

First day: The basics.

Squaring as a prototype of iteration. The superattractive role of infinity for polynomials. A closer look at Newton's method. Normal families and Montel's theorem.

Second day: The Julia set and the Fatou set.

Attractive and superatractive periodic orbits. The structure of the Julia set. Other type of non-caotic behaviour.

Third day: The role of the critical points.

The global structure of the Fatou set. Relation between the critical orbits and the conexity of the Julia set. The introduction of parameters in the picture.

This minicourse is a continuation of professor Poirier's exposition of basic complex dynamics. We focus on families of rational maps (more preciselly, polynomial families) and pass from the dynamical to the parameter plane, dealing with the interplay between them.

Our selected topics revolves around the (perhaps) main conjecture of complex dynamics: the density of hyperbolic components for families of rational maps. We present well known topics such as renormalization, invariant line fields, the Mañe-Sad-Sullivan paper, Yoccoz puzzles, etc.

All these topics are presented in the context of two polynomial families, one of them being that of quadratic polynomials, whose connectedness locus is the Mandelbrot set. The other one is a bicritical uniparametric family, object of study of the expositor's thesis.

En esta charla presentaré una relación entre los árboles de Hubbard y los árboles de DeMarco-McMullen (escaping trees). Dicha relación tiene lugar en el borde del lugar de conexidad en cierto conjunto de los polinomios de grado tres con un punto crítico de un periodo dado. Escribimos con Jan Kiwi una demostración de la conexidad de este conjunto de polinomios tal como conjeturado por John Milnor unos 30 años atrás. Les contaré como el trabajo sobres la relación entre esos árboles llevó unos años después a la demostración de la conjetura.

For each p>0 there is a family S_p of complex cubic maps with a marked critical orbit of period p. For each q>0 I will describe a dynamically defined tessellation of S_p. Each face of this tessellations is associated with one particular behavior for periodic orbits of period q.

The work described in this lecture is part of a general program in complex dynamics to understand parameter spaces of transcendental maps.

In all complex dynamical systems, the singular values control the stable periodic behavior. The singular values of rational functions are their critical values. Transcendental maps have a new kind of singularity, an “asymptotic value”: for example, 0,∞ for e^z and ±i for tan z. These functions belong to the relatively simple family F2 of transcendental maps with exactly two asymptotic values and no critical values. This family, up to affine conjugation, depends on two complex parameters.

In this lecture, we will begin by reviewing the structures of the parameter spaces of the exponential and tangent families which have been well studied. We will then describe recent work on two other slices of the full family F2. We will see how phenomena we observe for the tangent and the exponential families recur and combine in new ways.

In the moduli space of one variable complex cubic polynomials with a marked critical point, given any p>=1, we prove that the locus formed by polynomials with the marked critical point periodic of period $p$ is an irreducible curve. Our methods rely on techniques used to study one-complex-dimensional parameter spaces.

Sullivan’s celebrated no-wandering domains theorem for rational maps highlights a close connection or “dictionary” between holomorphic dynamics and Kleinian groups. The purpose of this talk is to highlight a new approach to the Sullivan dictionary, where for simplicity we focus on limit sets that generalize the Apollonian gasket. To each connected simple planar graph in the Riemann sphere, there is an associated circle packing by a theorem of Koebe-Andreev-Thurston. We give a dynamically natural way to associate both a Kleinian group and an anti-rational map to each such packing so that the limit and Julia sets are naturally identified. This identification enables the computation of the topological symmetry and quasisymmetry groups of the Julia set, and has led to new insights on the boundedness of deformation spaces and mate-ability.

Holomorphic correspondences are polynomial relations P(z,w)=0, which can be regarded as multi-valued self-maps of the Riemann sphere, this is implicit maps sending z to w. The iteration of such a multi-valued map generates a dynamical system on the Riemann sphere: dynamical system which generalises rational maps and finitely generated Kleinian groups. We consider a specific 1-(complex) parameter family of (2:2) correspondences F_a (introduced by S. Bullett and C. Penrose in 1994), which we describe dynamically. In particular, we show that for every parameter in a subset of the parameter plane called the connectedness locus and denoted by M_{\Gamma}, this family behaves as rational maps on a subset of the Riemann sphere and as the modular group on the complement: in other words, these correspondences are mating between the modular group and rational maps (in the family Per_1(1)). Moreover, we develop for this family of correspondences a complete dynamical theory which parallels the Douady-Hubbard theory of quadratic polynomials, and we show that M_{\Gamma} is homeomorphic to the parabolic Mandelbrot set M_1.

A study of real quadratic rational maps with real critical points up to an orientation preserving fractional linear change of variable. The moduli space consisting of all conjugacy classes of such maps is canonically diffeomorphic to S^1 x I.

Some regions of this moduli space correspond to dynamical behavior which is easy to describe, and others are more difficult. The description of the most difficult region will be based on the work of Khashayar Filom and Kevin Pilgrim. The talk will also briefly describe effective implementation of the Thurston pullback algorithm, and its behavior (or mis-behavior).

A rational map of degree at least 2 is hyperbolic if each of its critical points is attracted to an attracting cycle. The hyperbolic maps form an open subset in the space of rational maps and descends to an open subset in the corresponding moduli space of rational maps. Each component of this open subset is a hyperbolic component. In complex dynamics, an interesting question is to determine which types of hyperbolic components are bounded. In this talk, we study this problem in a well-known slice called Newton family. We prove that, in the moduli space of quartic Newton maps, a hyperbolic component is bounded if and only if all its root immediate basins have degree 2. Furthermore, for each unbounded hyperbolic component, we show that its boundary at infinity in the GIT-compactification is either a closed disk or a singleton. The proof is based on a convergence theorem of internal rays we establish for degenerate Newton sequences.

The local dynamics of a one-dimensional holomorphic germ tangent to the identity is described by the classical Leau-Fatou flower Theorem, showing how a pointed neighbourhood of the fixed point can be obtained as union of a finite number of forward or backward invariant open sets, the so-called petals of the Fatou flower, where the dynamics is conjugated to a translation in a half-plane.In this talk I will present what is known about generalizations of the Leau-Fatou flower Theorem to holomorphic germs tangent to the identity in several complex variables, where petals are replaced by parabolic curves. In particular, I will present a geometric proof of the fundamental results obtained by Écalle and Hakim on the existence of parabolic curves. This approach allows to give asymptotic expansions for the parametrization of parabolic curves for tangent to the identity holomorphic endomorphisms in a given neighbourhood of the fixed point

We will explain, with some examples, several ways to decribe one parameter slices of parameter spaces of rational maps.

Como consecuencia inmediata del Teorema de Poincaré-Bendixson, sabemos que las singularidades y las órbitas periódicas son los únicos conjuntos minimales de un campo de vectores en el plano bidimensional real. Aunque este resultado no tiene ningún paralelo en dimensiones mayores, en esta charla discutiremos una versión local para el caso de campos holomorfos cerca de un punto singular en el plano complejo bidimensional.

The finite Blaschke products are rational functions on the Riemann sphere that preserve the unit circle. Generally useful in studying the dynamics of polynomials; however, with a dynamic richness of its own.

The purpose of this talk is to explore some dynamical aspects of a Blaschke product family depending on a complex parameter, with a single critical point (cubic type) on the circle, critical value (the parameter), and two fixed super-atractors at zero and infinity. The variation of the critical value determines in the dynamical plane, the connectivity of the Julia sets and in the parameter plane, the existence of escape components, that is, parameters for which the critical point escapes by iteration to zero or infinity. Furthermore, we define the non-escape locus (Blaschkebrot) for the Blaschke family, where numerical experiments suggest the presence of baby cubibrots.

The notion of core entropy was introduced by W. Thurston by taking the entropy of the restriction of a complex quadratic polynomial to its Hubbard tree. This function varies wildly as the parameter varies, reflecting the topological complexity of the Mandelbrot set. Moreover, Thurston also defined the master teapot, a fractal set obtained by consideringfor each postcritically finite real quadratic polynomial the Galois conjugates of the entropy.

In the talk, we will discuss generalizations of this fractal from real to complex polynomials. In particular, we will define a master teapot for each vein the Mandelbrot set, discuss continuity properties of the core entropy, and use it to prove geometric properties of the master teapot.

Given a biholomorphism F with a fixed point from C^n to itself that admits a formal invariant curve , we give conditions that guarantee that there exists either a periodic curve, or a finite family of stable manifolds asymptotic to the formal curve. This generalizes the result on two dimensions proven by Lopez-Hernanz, Raissy, Ribon and Sanz-Sanchez.