We will study perturbations of the polynomials $P_λ$, of the form $$Q_λ= P_λ+Q$$ in the space of centered monic polynomials, where $P_λ$ is the polynomial family defined by $$P_λ(z)=\fracλ{p}z^p+(1-\fracλ{p}) z ^{p+1}$$ with $λ\in Λ= \{z: |z-1| <1\}$, studied by Benedetto, who showed that for a dense set of parameters, the polynomials $P_λ$ have a wandering disc contained in the filled Julia set.
We will show an analogous result for the family $Q_λ$, obtaining the following consequence:
The polynomials $P_λ$ belong to $\bar{\mathrm{E}}_{p+1}$ where $\mathrm{E}_{p+1}$ denotes the set of polynomials that have a wandering disc in the filled Julia set, in the space of centered monic polynomials of degree $p+1$.
This dissertation entitled "Fatou's No Wandering Domain Conjecture : Sullivan's Proof" contains a comprehensive study of the Dynamical Partition of the Riemann Sphere and a complementary analysis of the work done by Dennis Sullivan.
We construct polynomial automorphisms with wandering Fatou components. The four-dimensional automorphisms $H$ lie in a one-parameter family, depending on the parameter $δ\in \mathbb C \setminus \{0\}$, and as $δ\rightarrow 0$ the automorphisms degenerate to the two-dimensional polynomial map $P$ constructed in Astorg et al (Ann. of Math., 2016). Our main result states that if $P$ has a wandering domain, then $H$ does too for $δ$ sufficiently small.
We prove that there exist three transcendental entire functions that have infinite number of domains which lie in the wandering component of each of these functions and their composites. This result is a generalization of the result of Dinesh Kumar, Gopal Datt and Sanjay Kumar. In particular, they proved that there exist two transcendental entire functions that have infinite number of domains which lie in the wandering components of each of these functions and their composites.
We study the dynamics of polynomials with coefficients in a non-Archimedean field $K,$ where $K$ is a field containing a dense subset of algebraic elements over a discrete valued field $k.$ We prove that every wandering Fatou component is contained in the basin of a periodic orbit. We obtain a complete description of the new Julia set points that appear when passing from $K$ to the Berkovich line over $K$. We give a dynamical characterization of polynomials having algebraic Julia sets. More precisely, we establish that a polynomial with algebraic coefficients has algebraic Julia set if every c…
We show a partial generalization of Sullivan's non-wandering domain theorem in complex dimension two. More precisely, we show the non-existence of wandering Fatou components for polynomial skew products of $ \mathbb{C}^2$ with an invariant attracting fiber, under the assumption that the multiplier $ λ$ is small. We actually show a stronger result, namely that every forward orbit of any vertical Fatou disk intersects a bulging Fatou component.
We show that there exist polynomial endomorphisms of C^2, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of P^2(C). We also find real examples with wandering domains in R^2. The proof is based on parabolic implosion techniques, and is based on an original idea of M. Lyubich.
Little is known about the existence of wandering Fatou components for rational maps in two complex variables. In 2003 Lilov proved the non-existence of wandering Fatou components for polynomial skew-products in the neighborhood of an invariant super-attracting fiber. In fact Lilov proved a stronger result, namely that the forward orbit of any vertical disk must intersect a fattened Fatou component of the invariant fiber. Naturally the next class of maps to study are polynomial skew-products with an invariant attracting (but not super-attracting) fiber. Here we show that Lilov's stronger result…
We prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates $U_n$ of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values $p_n$ such that ${\rm dist}(p_n,\partial U_n)\to 0$ as $n\to \infty$. We also prove that if $U_n \cap P(f)=\emptyset$ and the postsingular set of $f$ l…
We prove that there exists a non-trivial transcendental semigroup S such that the wandering (or pre-periodic or periodic) components of Fatou set F(S) has at least a simply connected domain D.
Iteration of the function $f_λ(z)=λ+ z+\tan z, z \in \mathbb{C}$ is investigated in this article. It is proved that for every $λ$, the Fatou set of $f_λ$ has a completely invariant Baker domain $B$; we call it the primary Fatou component. The rest of the results deals with $f_λ$ when it is topologically hyperbolic. For all real $λ$ or $λ$ such that $ λ=πk +i λ_2$ for some integer $k$ and $0 < λ_2<1$, the only other Fatou component is shown to be another completely invariant Baker domain. It is proved that if $|2+λ^2|<1$, then the Fatou set is the union of $B$ and infinitely many invariant attr…
We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004 the non-existence of wandering domains near a super-attracting invariant fiber was shown in [8]. In 2014 it was shown in [1] that wandering domains can exist near a parabolic invariant fiber. In [9] the geometrically-attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew products; this class contains the maps studied in [9]. Using expansion properties on the Julia set in the invariant f…
We study the iteration of transcendental self-maps of $\mathbb{C}^*:=\mathbb{C}\setminus \{0\}$, that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a suffic…
We mainly generalize the notion of abelian transcendental semigroup to nearly abelian transcendental semigroup. We prove that Fatou set, Julia set and escaping set of nearly abelian transcendental semigroup are completely invariant. We investigate no wandering domain theorem in such a transcendental semigroup. We also obtain results on a complete generalization of the classification of periodic Fatou components.
A classification of the periodic components of the Fatou set of $p$-adic rational maps. Each such periodic component is either an immediate attracting basin or an open affinoid, where the dynamics is quasi-periodic (the $p$-adic analogues of Siegel discs and Herman rings.) The main tool is a fixed point theorem for connected subsets of Berkovich's projective line (or p-adic hyperbolic space).
Let $P$ be a polynomial of degree $d$ with Julia set $J_P$. Let $\widetilde N$ be the number of non-repelling cycles of $P$. By the famous Fatou-Shishikura inequality $\widetilde N\le d-1$. The goal of the paper is to improve this bound. The new count includes \emph{wandering collections of non-precritical branch continua}, i.e., collections of continua or points $Q_i\subset J_P$ \emph{all} of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of $\mathrm{eval}(Q_i)\ge 3$ external rays. Also, we relate individual cycles, which are either non-repelling or…
We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of $\mathbb{C}^m$, in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of bounded domains, namely for all bounded regular open sets $Ω\subset \mathbb{C}^m$ whose closure is polynomially convex. Our result gives in particular the first example of a bounded Fatou component with a smooth boundary in the category of holomorphic automorphisms.
In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component may be bounded even if the orbit of the component contains an infinite modulus annulus sequence and this case cannot happen when the maximal modulus of the meromorphic function is large enough. Therefore, we extend the related results for entire functions to ones for meromorphic functions with infinitely many poles. And we investigate the fast escaping Fatou components of meromorphic functions defined in [44] in terms of the Nevanlinna characteristic instead of the maximal modulus…
We investigate the description of Fatou components for polynomial skew-products in two complex variables. The non-existence of wandering domains near a super-attracting invariant fiber was shown in [L], and the geometrically-attracting case was studied in [PV] and [PS]. In [ABDPR] it was proven that wandering domains can exist near a parabolic invariant fiber. In this paper we study the remaining case, namely the dynamics near an elliptic invariant fiber. We prove that the two-dimensional Fatou components near the elliptic invariant fiber correspond exactly to the Fatou components of the restr…
While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of h…
The issue of whether an analytic function has wandering domains has long been of interest in complex dynamics. Sullivan proved in 1985 that rational maps do not have wandering domains. On the other hand, several transcendental entire functions have wandering domains. Using recent results on the relationship between Fatou components and the postsingular set, we find a new family of transcendental entire functions that does not have wandering domains. We also prove that the Julia set of a certain subfamily is the whole plane.
This paper is devoted to establish sufficient conditions under which a transcendental meromorphic function has no unbounded Fatou components and to extend some results for entire functions to meromorphic functions. Actually, we shall mainly discuss non-existence of unbounded wandering domains of a meromorphic function. The case for a composition of finitely many meromorphic function with at least one of them being transcendental can be also investigated in the argument of this paper.
Let $f$ be a transcendental entire function and $U$ be a Fatou component of $f$. We show that if $U$ is an escaping wandering domain of $f$, then most boundary points of $U$ (in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of $U$ are escaping, then $U$ is an escaping Fatou component. Some applications of these results are given; for example, if $I(f)$ is the escaping set of $f$, then $I(f)\cup\{\infty\}$ is connected.
The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental maps to consider. In this paper we find inspiration from the extensive work on the dynamics…
We show that wandering domains can exist in the Fatou set of a polynomial type quasiregular mapping of the plane. We also give an example of a quasiregular mapping of the plane, with an essential singularity at infinity, which has a sequence of wandering domains contained in a bounded part of the plane. This contrasts with the situation in the analytic case, where wandering domains are impossible for polynomials and, for transcendental entire functions, the existence of wandering domains in a bounded part of the plane has been an open problem for many years.
In 2001 E. Ghys, X. Gomez-Mont and J. Saludes defined the Fatou and Julia components of transversely holomorphic foliations on compact manifolds. It is a partition of the manifold in two saturated sets: the Fatou set which represents the non-chaotic part of the foliation and its complementary, the Julia set. Using the Brownian motion transverse to the foliation, it is proved in this paper that, if the foliation is taut and if F is a wandering component of the Fatou set, then almost every point of the topological boundary of F (almost for any harmonic measure on the boundary) is a limit point o…
This survey is an introduction to the classification of Fatou components in holomorphic dynamics. We start with the description of the Fatou and Julia sets for rational maps of the Riemann sphere, and finish with an account of the recent results on Fatou components for polynomial skew-products in complex dimension two, where we focus on the key steps in the construction giving the existence of a wandering domain for a polynomial endomorphism of $\mathbb{C}^2$.
We prove the existence of a locally dense set of real polynomial automorphisms of C 2 displaying a wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof is based on a geometric model for parameter families of surface real mappings. At a dense set of parameters, we show that the dynamics of the model displays a historical, high emergent, stable domain. We show that this model can be embedded into fami…
We describe conditions under which a multiply connected wandering domain of a transcendental meromorphic function with a finite number of poles must be a Baker wandering domain, and we discuss the possible eventual connectivity of Fatou components of transcendental meromorphic functions. We also show that if $f$ is meromorphic, $U$ is a bounded component of $F(f)$ and $V$ is the component of $F(f)$ such that $f(U)\subset V$, then $f$ maps each component of $\partial U$ onto a component of the boundary of $V$ in $\hat{\C}$. We give examples which show that our results are sharp; for example, we…
Let K be a non-archimedean field with residue field k, and suppose that k is not an algebraic extension of a finite field. We prove two results concerning wandering domains of rational functions f in K(z) and Rivera-Letelier's notion of nontrivial reduction. First, if f has nontrivial reduction, then assuming some simple hypotheses, we show that the Fatou set of f has wandering components by any of the usual definitions of such components. Second, we show that if k has characteristic zero and K is discretely valued, then the converse holds; that is, the existence of a wandering domain implies …
We consider a class of functions, denoted by K in this paper, which are meromorphic outside a compact and countable set B(f), investigated by A. Bolsch in his thesis in 1997. The set B(f) is the closure of isolated essential singularities. We review main definitions and properties of the Fatou and Julia sets of functions in class K. It is studied the role of B(f) in this context. Following Eremenko it is defined escaping sets and we prove some results related to them. For instance, the dynamics of a function is extended to its singularities using escaping hairs. We give an example of an escapi…
The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou-Julia-Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.
We discuss the dynamics of semigroups of transcendental entire functions using Fatou-Julia theory and provide a condition for the complete invariance of escaping set and Julia set of transcendental semigroups. Results regarding limit functions and postsingular set are also derived. In addition, classes of hyperbolic and postsingularly bounded transcendental semigroups are given. We also provide certain criterion for non existence of wandering domains of transcendental semigroups.
We extend a recent result on the existence of wandering domains of polynomial functions defined over the p-adic field C_p to any algebraically closed complete non-archimedean field C_K with residue characteristic p>0. We also prove that polynomials with wandering domains form a dense subset of a certain one-dimensional family of degree p+1 polynomials in C_K[z].
We consider the structure of substantially dissipative complex Hénon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points on the Julia set. Indeed, we prove the corresponding description of the Fatou set, namely that it consists of only finitely many components, each either attracting or parabolic periodic. In particular there are no rotation domains, and no wandering components. Moreover, we show that $J = J^\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycle…
The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function $f$ in any multiply connected wandering domain $U$ of $f$. By introducing a certain positive harmonic function $h$ in $U$, related to harmonic measure, we are able to give the first detailed description of this dynamical behaviour. Using this new technique, we show that, for sufficiently large $n$, the image domains $U_n=f^n(U)$ contain large annuli, $C_n$, and that the union of these annuli acts as …
We study the dynamics of a generic endomorphism $f$ of an Oka-Stein manifold $X$. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several descriptions of the Fatou set and the Julia set of $f$. In particular, we show that the Julia set is the derived set of the set of attracting periodic points of $f$ and that it is also the closure of the set of repelling periodic points of $f$. Among other results, we prove that $f$ is chaotic on the Julia set and that every periodic point of $f$ is hyperbolic. We a…
Let $f:\mathbb{C}^2\to \mathbb{C}^2$ be a polynomial skew product which leaves invariant an attracting vertical line $ L $. Assume moreover $f$ restricted to $L$ is non-uniformly hyperbolic, in the sense that $f$ restricted to $L$ satisfies one of the following conditions: 1. $f|_L$ satisfies Topological Collet-Eckmann and Weak Regularity conditions. 2. The Lyapunov exponent at every critical value point lying in the Julia set of $f|_L$ exist and is positive, and there is no parabolic cycle. Under one of the above conditions we show that the Fatou set in the basin of $L$ coincides with the uni…
Suppose that $f$ is a transcendental entire function, $V \subsetneq \mathbb{C}$ is a simply connected domain, and $U$ is a connected component of $f^{-1}(V)$. Using Riemann maps, we associate the map $f \colon U \to V$ to an inner function $g \colon \mathbb{D} \to \mathbb{D}$. It is straightforward to see that $g$ is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of $f$ in $V$ lie away from the boundary, there is a strong relationship between singularities of $g$ and accesses to inf…
For a transcendental entire function f, we study the set of points BU(f) whose iterates under f neither escape to infinity nor are bounded. We give new results on the connectedness properties of this set and show that, if U is a Fatou component that meets BU(f), then most boundary points of U (in the sense of harmonic measure) lie in BU(f). We prove this using a new result concerning the set of limit points of the iterates of f on the boundary of a wandering domain. Finally, we give some examples to illustrate our results.
A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the…
