We construct automorphisms of $\mathbb{C}^2$, and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank 1. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form $F(z,w)=(g(z,w),z)$ with $g(z,w):\mathbb{C}^2\rightarrow\mathbb{C}$ holomorphic.
We give an example of a transcendental entire function with a simply connected fast escaping Fatou component, but with no multiply connected Fatou components. We also give a new criterion for points to be in the fast escaping set.
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.
We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial $f$ have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge\exp(|z|^α)$ for some $α>0$ and all $z$ outside a set of finite Lebesgue measure.
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…
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 to what extent Fatou set, Julia set and escaping set of transcendental semigroup is respectively equal to the Fatou set, Julia set and escaping set of its subsemigroup. We define partial fundamental set and fundamental set of transcendental semigroup and on the basis of this set, we prove that Fatou set and escaping set of transcendental semigroup $ S $ are non-empty.
Let $f$ be Fatou's function, that is, $f(z)= z+1+e^{-z}$. We prove that the escaping set of $f$ has the structure of a `spider's web' and we show that this result implies that the non-escaping endpoints of the Julia set of $f$ together with infinity form a totally disconnected set. We also give a well-known transcendental entire function, due to Bergweiler, for which the escaping set is a spider's web and we point out that the same property holds for families of functions.
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.
In this paper, we prove that escaping set of transcendental semigroup is S-forward invariant. We also prove that if holomorphic semigroup is abelian, then Fatou set, Julia set and escaping set are S-completely invariant. We see certain cases and conditions that the holomorphic semigroup dynamics exhibits same dynamical behavior just like the classical complex dynamics. Frequently, we also examine certain amount of connection and contrast between classical complex dynamics and holomorphic semigroup dynamics.
We examine invariant nonrecurrent Fatou components of automorphisms of $\mathbb{C}^2$ in the case where all limit maps are constant. We show that except in special cases there cannot be more than one such limit map. We also briefly examine such Fatou components where the limit maps may be nonconstant. Lastly we present a few examples of such Fatou components.
We prove that there exists an automorphism of C^2 tangent to the identity with a domain of attraction to the origin, biholomorphic to the origin, along a degenerate characteristic direction.
We define commutator of a transcendental semigroup, and on the basis of this concept, we define conjugate semigroup. We prove that the conjugate semigroup is nearly abelian if and only if the given semigroup is nearly abelian. We also prove that image of each of escaping set, Julia set and Fatou set under commutator (affine complex conjugating maps) is equal to respectively escaping set, Julia set and Fatou set of conjugate semigroup. Finally, we prove that every element of the nearly abelian semigroup $ S $ can be written as the composition of an element from the set generated by the set of c…
We generalise a recent example by F. Bracci, J. Raissy and B. Stensønes to construct automorphisms of $\mathbb{C}^{d}$ admitting an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ and all attracting to a common boundary fixed point. These automorphisms can be chosen such that each Fatou component is invariant or such that the components are grouped into periodic cycles of any common period. We further show that no orbit in these attracting Fatou components can converge tangent to a complex submanifold, and that every sta…
We prove the existence of automorphisms of $\mathbb C^k$, $k\ge 2$, having an invariant, non-recurrent Fatou component biholomorphic to $\mathbb C \times (\mathbb C^\ast)^{k-1}$ which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. Such a Fatou component also avoids $k$ analytic discs intersecting transversally at the fixed point. As a corollary, we obtain a Runge copy of $\mathbb C \times (\mathbb C^\ast)^{k-1}$ in $\mathbb C^k$.
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.
Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible' under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$, we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$. In particular, we study functions for which $A(f)$, and each of its levels, has the structure of an `infinite …
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.
The family of exponential maps $f_a(z)= e^z+a$ is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set $J(f_a)$. When $a\in (-\infty,-1)$, and more generally when $a$ belongs to the Fatou set of $f_a$, it is known that $J(f_a)$ can be written as a union of "hairs" and "endpoints" of these hairs. In 1990, Mayer proved for $a\in (-\infty,-1)$ that, while the set of endpoints is totally separated, its union with infinity is a connected set. Recently, Alhabib and the second author extended this result to t…
We construct several new classes of transcendental entire functions, f, such that both the escaping set, I(f), and the fast escaping set, A(f), have a structure known as a spider's web. We show that some of these classes have a degree of stability under changes in the function. We show that new examples of functions for which I(f) and A(f) are spiders' webs can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders' webs to give new results concerning functions with no unbounded Fatou components.
We prove a form of the $\cos πρ$ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen's condition but…
We show that for any transcendental meromorphic function $f$ there is a point $z$ in the Julia set of $f$ such that the iterates $f^n(z)$ escape, that is, tend to $\infty$, arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which $f^n(z)$ tends to $\infty$ at a bounded rate, and establish the connections between these sets and the Julia set of $f$. To do this, we show that the iterates of $f$ satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a plane-filli…
We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in $\mathbb{R}^d$ is called hollow if it has a bounded complementary component. We show that for each $d \geq 2$ there exists a quasiregular map of transcendental type $f: \mathbb{R}^d \to \mathbb{R}^d$ with a quasi-Fatou component which is hollow. Suppose that $U$ is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if $U$ is bounded, then $U$ has many properties in common with a multiply connected Fatou component of a transcende…
We study the family of singular perturbations of Blaschke products $B_{a,λ}(z)=z^3\frac{z-a}{1-\overline{a}z}+\fracλ{z^2}$. We analyse how the connectivity of the Fatou components varies as we move continuously the parameter $λ$. We prove that all possible escaping configurations of the critical point $c_-(a,λ)$ take place within the parameter space. In particular, we prove that there are maps $B_{a,λ}$ which have Fatou components of arbitrarily large finite connectivity within their dynamical planes.
Let $f$ be a transcendental entire function and let $I(f)$ denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, $I(f)$ is connected. In particular, we show that $I(f)$ is connected if $f$ has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko's conjecture that $I(f)$ has no bounded components is true. We also give a new criterion related to $I(f)$ which is sufficient to ensure that $f$ has no unbounded Fatou components.
The fast escaping set, A(f), of a transcendental entire function f has begun to play a key role in transcendental dynamics. In many cases A(f) has the structure of a spider's web, which contains a sequence of fundamental loops. We investigate the structure of these fundamental loops for functions with a multiply connected Fatou component, and show that there exist transcendental entire functions for which some fundamental loops are analytic curves and approximately circles, while others are geometrically highly distorted. We do this by introducing a real-valued function which measures the rate…
We consider rational surface automorphisms with positive entropy. A Fatou component is said to be a rotation domain if the automorphism induces a torus action on it. Here we construct a rational surface automorphism with positive entropy with the following property: it has a rotation domain which contains both a curve of fixed points and isolated fixed points. This Fatou component cannot be imbedded into complex euclidean space, so we introduce a global linear model space and show that it can be globally linearized in this model.
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.
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 study basins of attraction of automorphisms of $\CC^2$ tangent to the identity that fix both axes. Our main result is that, if a well known conjecture about automorphisms of $\CC^*\times\CC^*$ holds, then there are no basins of attraction associated to the non-degenerate characteristic directions (in the sense of Hakim), and therefore we cannot find a Fatou-Bieberbach domain that does not intersect both axis with this method.
Let $f$ be an entire function with the form $f(z)=P(e^z)/e^z$, where $P$ is a polynomial with degree at least $2$ and $P(0)\neq 0$. We prove that the area of the complement of the fast escaping set (hence the Fatou set) of $f$ in a horizontal strip of width $2π$ is finite. In particular, the corresponding result can be applied to the sine family $α\sin(z+β)$, where $α\neq 0$ and $β\in\mathbb{C}$.
On automorphisms of super-level sets of Green's functions of Hénon maps
Indian Institute of Science Education and Research Mohali（印度科学教育研究所穆扎法拉布尔）；Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore Centre（印度统计学院班加罗尔中心理论与统计数学单元）
Let $f$ and $g$ be commuting meromorphic functions with finitely many poles. By studying the behaviour of Fatou components under this commuting relation, we prove that $f$ and $g$ have the same Julia set whenever $f$ and $g$ have no simply connected fast-escaping wandering domains. By combining this with a recent result of Tsantaris', we obtain the strongest statement (to date) regarding the Julia sets of commuting meromorphic functions. In order to highlight the difference to the entire case, we show that transcendental meromorphic functions with finitely many poles have orbits that alternate…
We show the existence of automorphisms $F$ of $\mathbb{C}^{2}$ with a non-recurrent Fatou component $Ω$ biholomorphic to $\mathbb{C}\times\mathbb{C}^{*}$ that is the basin of attraction to an invariant entire curve on which $F$ acts as an irrational rotation. We further show that the biholomorphism $Ω\to\mathbb{C}\times\mathbb{C}^{*}$ can be chosen such that it conjugates $F$ to a translation $(z,w)\mapsto(z+1,w)$, making $Ω$ a parabolic cylinder as recently defined by L.~Boc Thaler, F.~Bracci and H.~Peters. $F$ and $Ω$ are obtained by blowing up a fixed point of an automorphism of $\mathbb{C}…
The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties…
There are several classes of transcendental entire functions for which the Julia set consists of an uncountable union of disjoint curves each of which joins a finite endpoint to infinity. Many authors have studied the topological properties of this set of finite endpoints. It was recently shown that, for certain functions in the exponential family, there is a strong dichotomy between the topological properties of the set of endpoints which escape and those of the set of endpoints which do not escape. In this paper, we show that this result holds for large families of functions in the Eremenko-…
We introduce the concept of escaping set for semigroups of transcendental entire functions using Fatou-Julia theory. Several results of the escaping set associated with the iteration of one transcendental entire function have been extended to transcendental semigroups. We also investigate the properties of escaping sets for conjugate semigroups and abelian transcendental semigroups. Several classes of transcendental semigroups for which Eremenko's conjectures hold have been provided.
This paper consists of tow parts. One is to study the existence of a point $a$ in the intersection of Julia set and escaping set such that $\arg z=θ$ is a singular direction if $θ$ is a limit point of $\{\arg f^n(a)\}$ under some growth condition of a meromorphic function. The other is to study the connection between the Fatou set and singular direction. We prove that the absent of singular direction deduces the non-existence of annuli in the Fatou set.
A {\sl parabolic cylinder} is an invariant, non-recurrent Fatou component $Ω$ of an automorphism $F$ of $\mathbb C^2$ satisfying: (1) The closure of the $ω$-limit set of $F$ on $Ω$ contains an isolated fixed point, (2) there exists a univalent map $Φ$ from $Ω$ into $\mathbb C^2$ conjugating $F$ to the translation $(z,w) \mapsto (z+1, w)$, and (3) every limit map of $\{F^{\circ n}\}$ on $Ω$ has one-dimensional image. In this paper we prove the existence of parabolic cylinders for an explicit class of maps, and show that examples in this class can be constructed as compositions of shears and ove…
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 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…
We construct an entire function in the Eremenko-Lyubich class $\B$ whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in $\B$, in particular those of finite order, every escaping point can be connected to $\infty$ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.
We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metr…
The goal of this article is two fold. First, using transcendental shift-like automorphisms of C^k, k > 2 we construct two examples of non-degenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of C^k, k > 2 whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of Dixon-Esterle in C^2. The second example shows the existence of a Fatou--Bieberbach domain in C^k,k > 2 that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay-Rudi…
The purpose of this article is to explore a few properties of polynomial shift-like automorphisms of $\mathbb{C}^k.$ We first prove that a $ν-$shift-like polynomial map (say $S_a$) degenerates essentially to a polynomial map in $ν-$dimensions as $a \to 0.$ Secondly, we show that a shift-like map obtained by perturbing a hyperbolic polynomial (i.e., $S_a$, where $|a|$ is sufficiently small) has finitely many Fatou components, consisting of basins of attraction of periodic points and the component at infinity.
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…
The purpose of this paper is to present several examples of non--autonomous basins of attraction that arise from sequences of automorphisms of $\mathbb C^k$. In the first part, we prove that the non-autonomous basin of attraction arising from a pair of automorphisms of $\mathbb C^2$ of a prescribed form is biholomorphic to $\mathbb C^2$. This, in particular, provides a partial answer to a question raised in connection with Bedford's Conjecture about uniformizing stable manifolds. In the second part, we describe three examples of Short $\mathbb C^k$'s with specified properties. First, we show t…
The polynomial automorphisms of the affine plane have been studied a lot: if $f$ is such an automorphism, then either $f$ preserves a rational fibration, has an uncountable centralizer and its first dynamical degree equals $1$, or $f$ preserves no rational curves, has a countable centralizer and its first dynamical degree is $>1$. In higher dimensions there is no such description. In this article we study a family $(Ψ_α)_α$ of polynomial automorphisms of $\mathbb{C}^3$. We show that the first dynamical degree of $Ψ_α$ is $>1$, that $Ψ_α$ preserves a unique rational fibration and has an uncount…
