P. Domínguez, P. Makienko, G. Sienra
If $f$ is a transcendental entire function with only algebraic singularities we calculate the Ruelle operator of $f$. Moreover, we prove both (i) if $f$ has a summable critical point, then $f$ is not structurally stable under certain topological conditions and (ii) if all critical points of $f$ belonging to Julia set are summable, then there exists no invariant lines fields in the Julia set.
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We prove that entire transcendental holomorphic functions with an omitted value have infinite entropy. A proof for general transcendental entire functions will be given in an upcoming paper.
We exhibit an analytic family of hyperbolic, even disjoint type, entire functions for which the hyperbolic dimension does not vary analytically. Additionally we answer several questions in thermodynamic formalism of entire functions such as the existence of a hyperbolic entire function without conformal measure that is supported on the radial Julia set.
This text provides an overview of the (geometric) thermodynamic formalism for transcendental meromorphic and entire functions with particular emphasis on geometric/fractal aspects such as Bowen's Formula expressing the hyperbolic dimension as a unique zero of a pressure function and the behavior of the latter when the transcendental functions vary in an analytic family.
Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ ϕ= ψ\circ f$, where $ϕ,ψ:\C\to\C$ are affine). We show that the escaping sets of $f$ and $g$ have the same Hausdorff dimension. Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equal to one.
In this paper, we have investigated the Bungee set of composition of two transcendental entire functions. We have provided a class of permutable entire functions for which their Bungee sets are equal. Moreover, we have obtained a result on permutable entire functions for which the escaping set of the composite entire function equals the union of the escaping sets of the two functions. In addition, we establish important relations between the Bungee set, Escaping set and Filled Julia set of composite entire functions with that of its individual functions.
We study how the orbits of the singularities of the inverse of a meromorphic function prescribe the dynamics on its Julia set, at least up to a set of (Lebesgue) measure zero. We concentrate on a family of entire transcendental functions with only finitely many singularities of the inverse, counting multiplicity, all of which either escape exponentially fast or are pre-periodic. For these functions we are able to decide, whether the function is recurrent of not. In the case that the Julia set is not the entire plane we also obtain estimates for the measure of the Julia set.
We partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest. It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire functi…
Local connectivity of Julia sets of some transcendental entire functions with Siegel disks
本文利用拟Blaschke乘积长迭代的弱扩张性质，证明了一类具有有界型Siegel盘的超越整函数Julia集局部连通，并给出正弦函数实例及含渐近值情形的存在性结果。
We show that, if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider's web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider's web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider's web.
We prove that all entire transcendental entire functions have infinite topological entropy.
We study the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some conditions for connectivity of the Julia set of the transcendental semigroups. We also study finitely generated transcendental semigroups, abelian transcendental semigroups and limit functions of transcendental semigroups on its invariant Fatou components.
We consider the dynamics associated with an arbitrary semigroup of transcendental entire functions. Fatou-Julia theory is used to investigate the dynamics of these semigroups. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroup case. We also investigate the dynamics of conjugate semigroups, Abelian transcendental semigroups and wandering and Baker domains of transcendental semigroups.
We give examples of transcendental entire maps over $\mathbb{C}_p$ having an oscillating wandering Fatou component.
The notion of nearly abelian rational semigroup was introduced by Hinkannen and Martin. In this paper, we have introduced the notion of nearly abelian transcendental semigroup. We have extended the results of nearly abelian rational semigroups to the best possible class of nearly abelian transcendental semigroups. We have given a class of functions which nearly permute with a given transcendental entire function. In addition, a relation between the postsingular set of composition of two entire functions with that of individual functions is obtained.
Results on the postsingular set of compositions of transcendental entire functions
研究可交换超越整函数\(f\)与\(g\)（\(g = af^p + b\)）动力学，探讨奇异值和后奇异集在复合下性质，表明若一个函数具有某些性质，另一个及其复合也有，还得出关于超越半群的结果。
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.
We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.
Geometrically finite transcendental entire functions
本文引入“温顺函数”概念刻画超越整函数的拓扑动力学，证明一大类具有界临界性的几何有限超越整函数满足该性质，类比了多项式Julia集局部连通性的经典结果并推广了前期工作。
Deformations on Entire Transcendental Functions
研究整超越函数上 Baker 域的捏合形变收敛性，通过定义 Baker 叶状结构给出收敛与发散条件，并证明存在游荡域与后奇异集保持正欧氏距离的整函数。
In this article, we present a landing theorem for periodic dynamic rays for transcendental entire maps which have bounded post-singular sets, by using standard hyperbolic geometry results.
We use Bowen's definition of topological entropy and Ahlfors five islands theorem, as well as the theory of polynomial-like mappings, to show that the topological entropy of any entire transcendental function is infinity. In addition the entropy is concentrated on the Julia set for each meromorphic function which has no wandering domains.
We prove density of hyperbolicity in spaces of (i) real transcendental entire functions, bounded on the real line, whose singular set is finite and real and (ii) transcendental self-maps of the punctured plane which preserve the circle and whose singular set (apart from zero and infinity) is contained in the circle. In particular, we prove density of hyperbolicity in the famous Arnol'd family of circle maps and its generalizations, and solve a number of other open problems for these functions, including three conjectures by de Melo, Salomão and Vargas. We also prove density of (real) hyperboli…
The escaping set in transcendental dynamics
本文探讨超越函数逃逸集的定义、作用及其在动力学中的核心地位，总结主要研究成果并指出开放性问题。
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.
I show that the dynamical determinant, associated to an Anosov diffeomorphism, is the Fredholm determinant of the corresponding Ruelle-Perron-Frobenius transfer operator acting on appropriate Banach spaces. As a consequence it follows, for example, that the zeroes of the dynamical determinant describe the eigenvalues of the transfer operator and the Ruelle resonances and that, for $\Co^\infty$ Anosov diffeomorphisms, the dynamical determinant is an entire function.
Baker proved that for transcendental entire functions there is at most one completely invariant component of the Fatou set. It was observed by Julien Duval that there is a missing case in Baker's proof. In this article we follow Baker's ideas and give some alternative arguments to establish the result.
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。
Sierpiński carpet Julia sets of transcendental entire functions
本文证明谢尔宾斯基地毯参数在有限型超越整函数族的双曲分支点集中稠密，并构造了具有此类 Julia 集且含吸引盆、抛物盆、Siegel 盘或游荡域的具体函数。
We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that this conjugacy is es…
We consider the dynamical properties of transcendental entire functions and their compositions. We give several conditions under which Fatou set of a transcendental entire function $f$ coincide with that of $f\circ g,$ where $g$ is another transcendental entire function. We also prove some result giving relationship between singular values of transcendental entire functions and their compositions.
In recent years, there has been significant progress in the understanding of the dynamics of transcendental entire functions with bounded postsingular set. In particular, for certain classes of such functions, a complete description of their topological dynamics in terms of a simpler model has been given inspired by methods from polynomial dynamics. In this paper, and for the first time, we give analogous results in cases when the postsingular set is unbounded. More specifically, we show that if $f$ is of finite order, has bounded criticality on its Julia set $J(f)$, and its singular set consi…
Inner functions associated to lifts of transcendental entire functions
本文提出一种通用方法，将一类作为“提升”的整函数的内函数计算归结为被提升函数的内函数计算，推广了Evdoridou、Rempe和Sixmith的主要定理。
Infinite-dimensional Thurston theory and transcendental dynamics II: classification of entire functions with escaping singular orbits
本文利用无限维 Teichmüller 空间上的迭代程序，将 Thurston 对有理函数的经典拓扑刻画推广至无限标记点情形，从而分类所有奇异值沿互不相交射线逃逸的多项式与指数复合超越整函数。
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.
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 provide the full theory of thermodynamic formalism for a very general collection of entire functions in class $\mathcal B$. This class overlaps with the collection of all entire functions for which thermodynamic formalism has been so far established and contains many new functions. The key point is that we introduce an integral means spectrum for logarithmic tracts which takes care of the fractal behavior of the boundary of the tract near infinity. It turns out that this spectrum behaves well as soon as the tracts have some sufficiently nice geometry which, for example, is the case for quas…
Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental Hénon maps offers the potential of combining ideas from transcendental dynamics in one variable, and the dynamics of polynomial Hénon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.
Let f be a hyperbolic transcendental entire function of finite order in the Eremenko-Lyubich class (or a finite composition of such maps), and suppose that f has a unique Fatou component. We show that the Julia set of $f$ is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if $f\in\B$ has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic, then the Julia set of f contains a Cantor bouquet. As part of …
We construct a family of transcendental entire functions whose Julia sets have packing dimension in $(1,2)$. These are the first examples where the computed packing dimension is not $1$ or $2$. Our construction will allow us further show that the set of packing dimensions attained is dense in the interval $(1,2)$, and that the Hausdorff dimension of the Julia sets can be made arbitrarily close to the corresponding packing dimension.
Let $f$ be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated to a singular orbit which cannot accumulate on any other non-repelling cycle. When $f$ has finitely many singular values this implies a refinement of the Fatou-Shishikura inequality. Our approach is combinatorial in the spirit of the approach used by [Ki00], [BCL+16] for polynomials.
Infinite-dimensional Thurston theory and transcendental dynamics III: entire functions with escaping singular orbits in the degenerate case
本研究针对多项式与指数复合的超越整函数，聚焦奇异值沿不交射线退化逃逸的情形，通过无穷维Teichmüller空间迭代（类比Thurston有理函数拓扑刻画），实现该类函数的分类。
A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this property, and this class has recently attracted much attention in complex dynamics. In the presence of escaping critical values, these curves break or split at (preimages of) critical points. In this paper, we develop combinatorial tools that allow us to provide a complete description of the escaping set of any criniferous function without asymptotic values on its…
We 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 disc. After normalisation this is a one parameter family $f_a$ with $a\in\mathbb{C}^*$ which includes the semi-standard map $λze^z$ at $a=1$, approaches the exponential map when $a\to0$ and a quadratic polynomial when $a\to\infty$. We investigate the stable components of the parameter plane (capture components and semi-hyperbolic components) and also some topological properties of the Siegel disc in terms of th…
Many authors have studied sets, associated with the dynamics of a transcendental entire function, which have the topological property of being a spider's web. In this paper we adapt the definition of a spider's web to the punctured plane. We give several characterisations of this topological structure, and study the connection with the usual spider's web in $\mathbb{C}$. We show that there are many transcendental self-maps of $\mathbb{C}^*$ for which the Julia set is such a spider's web, and we construct a transcendental self-map of $\mathbb{C}^*$ for which the escaping set $I(f)$ has this str…
Transcendental correspondences: when Fuchsian groups take over basins of entire maps
本文系统研究超越整函数与富克斯群共形组合产生的\((\infty:\infty)\)全纯对应，结合整函数盆域外动力学与群作用，证明其为特定复合，当整函数奇异值有限时，相关亚纯函数奇异值也有限且线复形可明确描述。
The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire functions. Here the Julia set is defined as the set of all points such that complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.
On Dynamics of the Bungee set and the Filled Julia set of a Transcendental Semigroup
本文利用Fatou-Julia理论定义超越半群的bungee集与填充Julia集，推广单函数结果，给出包含条件并探讨逃逸集及共轭、交换半群的新性质。
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…
We define two classes of topological infinite degree covering maps modeled on two families of transcendental holomorphic maps. The first, which we call exponential maps of type $(p,q)$, are branched covers and is modeled on transcendental entire maps of the form $P e^{Q}$, where $P$ and $Q$ are polynomials of degrees $p$ and $q$. The second is the class of universal covering maps from the plane to the sphere with two removed points modeled on transcendental meromorphic maps with two asymptotic values. The problem we address is to give a combinatorial characterization of the holomorphic maps co…
Thermodynamic formalism for entire transcendental maps with hyperbolic Baker Domains
本文为具有双曲Baker域的整超越映射建立了热力学形式，证明了共形测度的存在唯一性，并得出Hausdorff维数是压力函数在$t>1$时的唯一零点。
Let $f$ be a transcendental entire function. The fast escaping set, $A(f)$, plays a key role in transcendental dynamics. The quite fast escaping set, $Q(f)$, defined by an apparently weaker condition is equal to $A(f)$ under certain conditions. Here we introduce $Q_2(f)$ defined by what appears to be an even weaker condition. Using a new regularity condition we show that functions of finite order and positive lower order satisfy $Q_2(f)=A(f)$. We also show that the finite composition of such functions satisfies $Q_2(f)=A(f)$. Finally, we construct a function for which $Q_2(f) \neq Q(f)= A(f)$.
We identify a class of hyperbolic transcendental entire maps and we prove that some of its elements generate a class of potentials for which exhibit a conformal and invariant probability Gibbs measure. The methods and techniques from the thermodynamic formalism can be extended to this class of potentials. To complement this study we highlight that the dynamics of such a map on some subset of the Julia set is conjugated to the shift map over a code space with countable alphabet and the euclidean metric on the complex plane induces a metric on the symbolic space which is not compatible with the …
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 show that for a transcendental entire function the set of points whose orbit under iteration is bounded can have arbitrarily small positive Hausdorff dimension.
Transcendental Julia Sets of Minimal Hausdorff Dimension
本文证明存在超越整函数，其朱利亚集豪斯多夫维数为1且每个法图分支无限连通，并存在单点补分支，解答了Rippon和Stallard的问题，方法基于拟共形手术。
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…
In this paper, we study fast escaping set of transcendental semigroup. We discuss some the structure and properties of fast escaping set of transcendental semigroup. We also see how far the classical theory of fast escaping set of transcendental entire function applies to general settings of transcendental semigroups and what new phenomena can occur.
Suppose that $f$ is a transcendental entire function. In 2014, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected, and an example a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class. It is known that for many transcendental entire functions the escaping set…
We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.
We consider the family of transcendental entire maps given by $f_a(z)=a(z-(1-a))\exp(z+a)$ where $a$ is a complex parameter. Every map has a superattracting fixed point at $z=-a$ and an asymptotic value at $z=0$. For $a>1$ the Julia set of $f_a$ is known to be homeomorphic to the Sierpiński universal curve, thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of non-landing hairs with prescribed combinatorics embedded in the Julia set for al…
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