Mareike Wolff
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.
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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 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 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.
Escaping Fatou components with disjoint hyperbolic limit sets
本文构造了二维复空间的自同构，其逃逸Fatou分支循环上恰有两个秩为1的极限函数，且对应极限集为无穷远直线上两个不交的双曲子集。
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 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…
Automorphisms of $\mathbb{C}^2$ with cycles of escaping Fatou components with hyperbolic limit sets
研究形如 $F(z,w)=(e^{-z^m}+ \delta e^{\frac{2 \pi}{m}i}\\, w\\,,\\,z)$ 的 $\mathbb{C}^2$ 非多项式自同构的稳定动力学，揭示了奇偶参数 $m$ 下逃逸Fatou分支循环的周期与双曲极限集特征。
We give criteria for the escaping set and the Julia set of an entire function to have positive measure. The results are applied to Poincaré functions of semihyperbolic polynomials and to the Weierstraß $σ$-function.
We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated points and that most zeros are located in neighborhoods of finitely many rays. We apply the result to solutions of linear differential equations.
Non-bulging Fatou components for transcendental skew-products
本文研究了斜积中不变纤维上逃逸或振荡福图组件的鼓起现象，探讨扰动对鼓起的影响。证明了适当扰动下，任何轨道无界组件非鼓起，而足够良好的扰动可使吸引纤维上的组件鼓起。
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 …
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…
For a transcendental entire function $f$ of finite order in the Eremenko-Lyubich class $\mathcal{B}$, we give conditions under which the Lebesgue measure of the escaping set $\mathcal{I}(f)$ of $f$ is zero. This is inspired by the recent work of Aspenberg and Bergweiler, in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko-Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to the case of infini…
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…
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.
Slowly recurrent Collet-Eckmann maps with non-empty Fatou set
该研究针对Julia集非全球面、临界点慢回复的有理Collet-Eckmann映射，证明其在固定临界点阶数的族中是双曲映射的Lebesgue密度点，单临界点情形下在任意次数≥2的有理映射全空间中也成立。
Fatou Components of Attracting Skew Products: From Polynomial Classification to Transcendental Complexity
该研究对全纯斜积的法图分支做精细分类，刻画多项式斜积无穷远吸引域的鼓胀性条件，在超越情形构造部分鼓胀型分支并证明鼓胀性依赖纤维动力学与扰动的相互作用。
Eremenko's conjecture, wandering Lakes of Wada, and maverick points
本文开发了一种将复平面全闭子集实现为整函数游荡集的通用技术，构造了 Eremenko 猜想的反例，并解决了关于 Fatou 分支共享边界、非逃逸点测度及单连通分支闭包补集等多个复动力学开放问题。
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.
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 $β\in\mathbb{C}$.
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.
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 study the dynamical behaviour of points in the boundaries of simply connected invariant Baker domains $U$ of meromorphic maps $f$ with a finite degree on $U$. We prove that if $f|_U$ is of hyperbolic or simply parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure escapes to infinity under iteration. On the contrary, if $f|_U$ is of doubly parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure has dense forward trajectory in the boundary of $U$, in particular the set of escaping points in the boundary of $U$ h…
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…
We consider the escaping parameters in the family $β\wp_Λ$, i.e. these parameters for which the orbits of critical values of $β\wp_Λ$ approach infinity, where $\wp_Λ$ is the Weierstrass function. Unlike to the exponential map the considered functions are ergodic. They admit a non-atomic, $σ$-finite, ergodic, conservative and invariant measure $μ$ absolutely continuous with respect to the Lebesgue measure. Under additional assumptions on the $\wp_Λ$-function we estimate from below the Hausdorff dimension of the set of escaping parameters in the family $β\wp_Λ$, and compare it with the Hausdorff…
A survey on permutable transcendental entire functions and their dynamics
综述1958年至2025年可交换超越整函数动力学成果，探讨其函数形式、动力学性质，分析基于点轨道动力学分类的子集关系及逃逸集分类，还讨论了相关集合的动力学关系与蹦极集和游荡域的情况。
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 study the dynamics of iterated cosine maps $E\colon z \mapsto ae^z+be^{-z},$ with $a,b \in \C\setminus \{0\}$. We show that the points which converge to infinity under iteration are organized in the form of rays and, as in the exponential family, every escaping point is either on one of these rays or the landing point of a unique ray. Thus we get a complete classification of the escaping points of the cosine family, confirming a conjecture of Eremenko in this case. We also get a particularly strong version of the ``dimension paradox'': the set of rays has Hausdorff dimension 1, while the se…
Polynomial Random Dynamical Systems with Complete Connections and the Probability of Tending to Infinity
本文研究了Riemann球面上的多项式随机动力系统，探讨了记忆依赖的下一多项式映射选择规则，证明了在Fatou集上概率局部常数，并在核Julia集为空时连续，同时引入了平稳平均逃逸概率。
A transcendental entire function f is called geometrically finite if the intersection of the set of singular values with the Fatou set is compact and the intersection of the postsingular set with the Julia set is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then the Fatou set of f is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles. Let z_0 be a repelling or parabolic periodic point of such a map f. We show that, if f has finite order, then there exists an injective cu…
Let f be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set F(f) and the postsingular set P(f) is compact and the intersection of the Julia set J(f) and P(f) is finite. Assume that no asymptotic value of f belongs to J(f) and that the local degree of f at all points in J(f) is bounded by some finite constant. We prove that there is a hyperbolic map g (of the form g(z)=f(bz) for some complex number b) with connected Fatou set such that f and g are semiconjugate on their Julia sets. Furthermore, we show that this semiconjugacy is a conjugacy when restricte…
Perturbations of exponential maps: Non-recurrent dynamics
该研究探讨指数族中非回归参数的扰动，证明其集合勒贝格测度为零，补充了Qiu 1994年关于逃逸参数的结论，还证实非回归参数可由双曲参数逼近。
On the boundary of an immediate attracting basin of a hyperbolic entire function
本文研究了有限阶超越整函数的吸引周期点，在附加假设下证明其 Fatou 集分量边界与逃逸集交集的 Hausdorff 维数为 1，推广了已有结果。
Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)|>R has N components for some R>0, then the order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpe…
Entire functions with Cantor bouquet Julia sets
本文研究超越整函数的Cantor bouquet Julia集，证明了criniferous性质与发丝结构的等价性，给出基于吸收集的新刻画，并在几何假设下证明head-start条件既是充分的也是必要的。
Computability of Julia sets for complex Hénon maps: The role of attracting and neutral cycles
该研究将复Hénon映射Julia集的可计算性从双曲情形扩展到含吸引盆地的非双曲情形，基于鞍点稳定流形的算法实现，同时指出中性动力学下高维存在部分类似一维的可计算性二分法。
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