9 research outputs found
On McMullen-like mappings
We introduce a generalization of the McMullen family
. In 1988, C. McMullen showed that the Julia
set of is a Cantor set of circles if and only if and
the simple critical values of belong to the trap door. We
generalize this behavior defining a McMullen-like mapping as a rational map
associated to a hyperbolic postcritically finite polynomial and a pole data
where we encode, basically, the location of every pole of and
the local degree at each pole. In the McMullen family, the polynomial is
and the pole data is the pole located at the
origin that maps to infinity with local degree . As in the McMullen family
, we can characterize a McMullen-like mapping using an arithmetic
condition depending only on the polynomial and the pole data .
We prove that the arithmetic condition is necessary using the theory of
Thurston's obstructions, and sufficient by quasiconformal surgery.Comment: 21 pages, 2 figure
Wandering domains for composition of entire functions
C. Bishop has constructed an example of an entire function f in
Eremenko-Lyubich class with at least two grand orbits of oscillating wandering
domains. In this paper we show that his example has exactly two such orbits,
that is, f has no unexpected wandering domains. We apply this result to the
classical problem of relating the Julia sets of composite functions with the
Julia set of its members. More precisely, we show the existence of two entire
maps f and g in Eremenko-Lyubich class such that the Fatou set of f compose
with g has a wandering domain, while all Fatou components of f or g are
preperiodic. This complements a result of A. Singh and results of W. Bergweiler
and A. Hinkkanen related to this problem.Comment: 21 pages, 3 figure
Wandering domains for composition of entire functions
C. Bishop in [Bis, Theorem 17.1] constructs an example of an entire function in class with at least two grand orbits of oscillating wandering domains. In this paper we show that his example has exactly two such orbits, that is, has no unexpected wandering domains. We apply this result to the classical problem of relating the Julia sets of composite functions with the Julia set of its members. More precisely, we show the existence of two entire maps and in class such that the Fatou set of has a wandering domain, while all Fatou components of or are preperiodic. This complements a result of A. Singh in [Sin03, Theorem 4] and results of W. Bergweiler and A.Hinkkanen in [BH99] related to this problem
Construction de fractions rationnelles à dynamique prescrite
In this thesis, we are interested in the existence criterions and the effective construction of rational maps with prescribed dynamics. We start by studying the same problem for some post-critically finite ramified coverings and we give a construction method from dynamical trees. Then we present a Thurston's theorem which provides a combinatorial characterization to go from the topological point of view to the analytical one. In particular, we generalize to non-post-critically finite maps a Levy's result which simplifies the Thurston's criterion in the polynomial case. We illustrate this generalization by a sufficient condition for existence of polynomials with a fixed Siegel disk of bounded type. Next we detail the construction by quasiconformal surgery of an example of non-post-critically finite rational map whose dynamics is described by a tree. More generally, we show that a result of Cui Guizhen and Tan Lei allows to construct a family of rational maps with disconnected Julia sets from some weighted Hubbard trees.Dans cette thèse, nous nous intéressons aux critères d'existence et à la construction effective de fractions rationnelles à dynamique prescrite. Nous commençons par étudier le même problème pour certains revêtements ramifiés post-critiquement finis et nous donnons une méthode de construction à partir de dynamiques d'arbres. Puis nous présentons un théorème de Thurston qui fournit une caractérisation combinatoire pour passer du cadre topologique au cadre analytique. En particulier, nous généralisons aux applications non post-critiquement finies un résultat de Levy qui simplifie le critère de Thurston dans le cas polynomial. Nous illustrons cette généralisation par une condition suffisante d'existence de polynômes ayant un disque de Siegel fixe de type borné. Ensuite nous détaillons la construction par chirurgie quasiconforme d'un exemple de fraction rationnelle non post-critiquement finie dont la dynamique est décrite par un arbre. Plus généralement, nous montrons qu'un résultat de Cui Guizhen et Tan Lei permet de construire une famille de fractions rationnelles à ensemble de Julia disconnexe à partir de certains arbres de Hubbard pondérés
Introduction to Iterated Monodromy Groups
The theory of iterated monodromy groups was developed by Nekrashevych. It is
a wonderful example of application of group theory in dynamical systems and, in
particular, in holomorphic dynamics. Iterated monodromy groups encode in a
computationally efficient way combinatorial information about any dynamical
system induced by a post-critically finite branched covering. Their power was
illustrated by a solution of the Hubbard Twisted Rabbit Problem given by
Bartholdi and Nekrashevych.
These notes attempt to introduce this theory for those who are familiar with
holomorphic dynamics but not with group theory. The aims are to give all
explanations needed to understand the main definition and to provide skills in
computing any iterated monodromy group efficiently. Moreover some explicit
links between iterated monodromy groups and holomorphic dynamics are detailed.
In particular, combinatorial equivalence classes and matings of polynomials are
discussed.Comment: 38 pages, 20 figures. Published in Ann. Fac. Sci. Toulous
Wandering domains for composition of entire functions
C. Bishop in [Bis, Theorem 17.1] constructs an example of an entire function in class with at least two grand orbits of oscillating wandering domains. In this paper we show that his example has exactly two such orbits, that is, has no unexpected wandering domains. We apply this result to the classical problem of relating the Julia sets of composite functions with the Julia set of its members. More precisely, we show the existence of two entire maps and in class such that the Fatou set of has a wandering domain, while all Fatou components of or are preperiodic. This complements a result of A. Singh in [Sin03, Theorem 4] and results of W. Bergweiler and A.Hinkkanen in [BH99] related to this problem