Invariants en dinàmica complexa

Usant com a fil conductor el mètode de Newton per a polinomis complexos, aquesta lliçó pretén mostrar els diferents comportaments que poden tenir les òrbites d?un sistema dinàmic generat per la iteració d?una funció analítica del pla complex. Veurem com aquestes òrbites s?agrupen en conjunts invariants amb dinàmiques molt variades, separats per fronteres fractals amb propietats remarcables, tant topològiques com dinàmiques.Using Newton’s method for complex polynomials as a conducting theme, this lecture tries to show the possible asymptotic behaviours of orbits under iteration of holomorphic maps. We see how these orbits form invariant sets with different possible dynamics, separated by fractal boundaries with amazing topological and dynamical properties

Surgery on Herman rings of the complex standard family

We consider the standard family (or Arnold family) of circle maps given by f_{\alpha, \beta}(x)=x + \alpha + \beta \sin(x) \pmod{2\pi}, for x,\alpha\in [0,2\pi), \beta \in (0,1) and its complexification F_{\alpha,\beta}(z)=z e^{i\alpha} \exp [\frac12\beta(z-\frac{1}{z})]. If f_{\alpha,\beta} is analytically linearizable, there is a Herman ring around the unit circle in the dynamical plane of F_{\alpha,\beta}. Given an irrational rotation number \theta, the parameters (\alpha,\beta) such that f_{\alpha, \beta} has rotation number \theta form a curve T_\theta in the parameter plane. Using quasi-conformal surgery of the simplest type, we show that if \theta is a Brjuno number, the curve T_\theta can be parametrized real-analytically by the modulus of the Herman ring, from \beta=0 up to a point (\alpha_0,\beta_0) with \beta_0 \leq 1, for which the Herman ring collapses. Using a result of Herman and a construction in I. N. Baker and P. Domínguez (Complex Variables37 (1998), 67-98) we show that for a certain set of angles \theta \in \mathcal{B} \setminus \mathcal{H}, the point \beta_0 is strictly less than 1 and, moreover, the boundary of the Herman rings with the corresponding rotation number have two connected components which are quasi-circles, and do not contain any critical point. For rotation numbers of constant type, the boundary consists of two quasi-circles, each containing one of the two critical points of F_{\alpha, \beta}

Invariants en Dinàmica complexa

Usant com a fil conductor el mètode de Newton per a polinomis complexos, aquesta lliçó pretén mostrar els diferents comportaments que poden tenir les òrbites d'un sistema dinàmic generat per la iteració d'una funció analítica del pla complex. Veurem com aquestes òrbites s'agrupen en conjunts invariants amb dinàmiques molt variades, separats per fronteres fractals amb propietats remarcables, tant topològiques com dinàmiques

Deformations of entire functions with Baker domains

We consider entire transcendental functions $f$ with an invariant (or periodic) Baker domain $U$. First, we classify these domains into three types (hyperbolic, simply parabolic and doubly parabolic) according to the surface they induce when we take the quotient by the dynamics. Second, we study the space of quasiconformal deformations of an entire map with such a Baker domain by studying its Teichmuüller space. More precisely, we show that the dimension of this set is infinite if the Baker domain is hyperbolic or simply parabolic, and from this we deduce that the quasiconformal deformation space of $f$ is infinite dimensional. Finally, we prove that the function $f(z)=z+e^{-z}$, which possesses infinitely many invariant Baker domains, is rigid, i.e., any quasiconformal deformation of $f$ is affinely conjugate to $f$

Stable components in the parameter plane of transcendental functions of finite type

We study the parameter planes of certain one-dimensional, dynamically-de ned slices of holomorphic families of entire and meromorphic transcendental maps of nite type. Our planes are de ned by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call shell components, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, the virtual center, which plays the same role. For entire slices, the virtual center is always at in nity, while for meromorphic ones it maybe nite or in nite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected. Our dynamical plane results apply without the restriction of nite type

An entire transcendental family with a persistent Siegel disc

Altres ajuts: Consolider and European funds MCRTN-CT-2006-035651We study the class of entire transcendental maps of finite order with one critical point and one asymptotic value, which has exactly one finite pre-image, and having a persistent Siegel disk. After normalization this is a one parameter family fa with a ∈ C which includes the semistandard map λzez at a = 1, approaches the exponential map when a → 0 and a quadratic polynomial when a → ∞. We investigate the stable components of the parameter plane (capture components and semi-hyperbolic components) and also some topological properties of the Siegel disk in terms of the parameter

The Fine Structure of Herman Rings

We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3 (up to quasiconformal deformation). Shishikura's quasiconformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we transfer McMullen's results on the fine local geometry of Siegel disks to the Herman ring setting

El conjunt de Mandelbrot i altres plans de bifurcació

El conjunt de Mandelbrot va ser vist físicament per primera vegada en una pantalla cap al començament dels anys vuitanta. Aquest fet va obrir la possibilitat a l'experimentació numèrica i va renovar l'interès en les matemàtiques que hi ha al darrera d'aquesta intrigant i bella imatge, les quals havien estat en un punt mort des de feia gairebé vuitanta anys [1]. El ressorgir definitiu va tenir lloc molt poc després [15], amb la introducció d'una nova eina que ha demostrat ser fonamental: les funcions quasiconforme

Dynamic rays of bounded-type transcendental self-maps of the punctured plane

We study the escaping set of functions in the class B∗, that is, transcendental self-maps of C∗ for which the set of singular values is contained in a compact annulus of C∗ that separates zero from infinity. For functions in the class B∗, escaping points lie in their Julia set. If f is a composition of finite order transcendental self-maps of C∗ (and hence, in the class B∗), then we show that every escaping point of f can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every sequence e∈{0,∞}N0, we show that the escaping set of f contains a Cantor bouquet of curves that accumulate to the set {0,∞} according to e under iteration by f

The fine structure of Herman rings

We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3 (up to quasiconformal deformation). Shishikura's quasi-conformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we transfer McMullen's results on the fine local geometry of Siegel disks to the Herman ring setting