Entropy of Transcendental entire functions

Dynamics on the Wandering Components of the Fatou Set of Three Transcendental Entire Functions and Their Composites

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.

A complementary proof of Baker's theorem of completely invariant components for transcendental entire functions

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.

超越整函数的谢尔宾斯基地毯 Julia 集

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。

关于复合整函数与奇点的动力学

On dynamics of composite entire functions and singularities

研究了超越整函数及其合成的动力学性质。给出了超越整函数$f$的法图集与$f\circ g,$的Fatou集重合的几个条件，其中$g$是另一个超越整函数.证明了超越整函数的奇异值与其复合函数的奇异值之间的关系。

无限维 Thurston 理论与超越动力学 II：具有逃逸奇异轨道的整函数的分类

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。

Infinite entropy for transcendental entire functions with an omitted value

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.

Singular values and non-repelling cycles for entire transcendental maps

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.

无穷维Thurston理论与超越动力学III：退化情形下具有逃逸奇异轨道的整函数

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。

整个超自然家族都有一个持续的西格尔圆盘

An entire transcendental family with a persistent Siegel disc

研究了一类具有一个临界点和一个渐近值的有穷级超越映射，它具有一个有限原像，且具有持久Siegel圆.在归一化之后，这是具有$a\in\mathbb{C}^*$的单参数族$f_a$，其包括在$a=1$处的半标准映射$λze^z$，当$a\to0$时接近指数映射，并且当$a\to\infty$时接近二次多项式。我们研究了参数平面上的稳定分量（捕获分量和半双曲分量）以及Siegel圆盘在参数下的一些拓扑性质。

有界几何与某些超越整映射和亚纯映射的特征

Bounded Geometry and Characterization of Some Transcendental Entire and Meromorphic Maps

定义了两类拓扑无限度覆盖映射，它们是以两类超越全纯映射为模型的。第一个，我们称之为$(p,q)$型指数映射，是分支覆盖，并以$P e^{Q}$形式的超越整映射为模型，其中$P$和$Q$是次数为$p$和$q$的多项式。第二类是具有两个渐近值的超越亚纯映射上的从平面到球面的具有两个去点的泛覆盖映射。我们解决的问题是给出包含在这些类的后奇异集是有限的全纯映射的组合特征。本文的主要结果是：一个$(0,1)$型的后奇异有限拓扑指数映射或一个$(p,1)$型的后奇异有限拓扑指数映射或一个从平面到球面的后奇异有限泛覆叠映射映射组合等价于一个全纯同型映射当且仅当该映射有界几何.

Thermodynamic Formalism and Geometric Applications for Transcendental Meromorphic and Entire Functions

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.

Ruelle算子与超越整映射

Ruelle operator and transcendental entire maps

如果$f$是一个只有代数奇点的超越整函数，我们计算$f$的Ruelle算子。证明了：（i）若$f$有可和临界点，则在一定的拓扑条件下，$f$不是结构稳定的;（ii）若$f$的所有临界点都属于朱利亚集，则在朱利亚集中不存在不变线场.

具有双曲Baker域的整超越映射的热力学形式

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。

超越整函数逃逸集的豪斯多夫维数

Hausdorff dimensions of escaping sets of transcendental entire functions

设$f$和$g$是超越整函数，每个函数都有一个有界奇异值集，并假设$f$和$g$是仿射等价的（即，$g \circ ϕ= ψ\circ f$，其中$ϕ,ψ:\C\to\C$是仿射的）。证明了$f$和$g$的逃逸集具有相同的豪斯多夫维数.利用第二作者的一个结果，我们证明了存在一族超越整函数，其逃逸集的豪斯多夫维数等于1.

On the existence of a conformal and an absolutely continuous invariant measure for transcendental entire maps

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 …

Dynamics on Bungee set, Escaping set and Filled Julia set of Composite Transcendental Entire Functions

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.

Dynamics of transcendental Hénon maps III: Infinite entropy

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.

具有简单后奇异集的整超越函数的常返性

Recurrence of entire transcendental functions with simple post singular sets

研究了亚纯函数的逆函数的奇点轨道如何在其朱利亚集上定义动力学，至少直到一个Lebesgue测度为零的集合.我们集中在一个家庭的整个超越函数只有1000个奇异的逆，计数的多重性，所有这些要么逃逸指数快速或前周期。对于这些函数，我们能够决定函数是否是递归的。在朱利亚集不是整个平面的情况下，我们还得到了朱利亚集的测度的估计.

超越整函数的极大和非极大快速逃逸点

Maximally and non-maximally fast escaping points of transcendental entire functions

将超越整函数的快速逃逸集划分为极大快速逃逸集和非极大快速逃逸集。这些集合具有很强的动力学性质。证明了非极大快速逃逸集与朱利亚集的交集不为空。证明使用了一个新的覆盖结果的环，这是更广泛的兴趣。Rippon和Stallard证明了快速逃逸集没有有界分量。相反，通过研究一个函数认为哈代，我们给出了一个例子的超越整函数的最大和非最大快速逃逸集每个都有无数个单点组件。

具有Siegel盘的一类超越整函数Julia集的局部连通性

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。

蜘蛛网与超越整函数的locally connected朱利亚集

Spiders' webs and locally connected Julia sets of transcendental entire functions

我们表明，如果一个超越整函数的朱利亚集是局部连通的，那么它的形式的蜘蛛网的意义上定义的Rippon和Stallard。在相反的方向上，我们证明了蜘蛛网朱利亚集总是局部连通的一个稠密子集的掩埋点。我们还表明，一组埋点（剩余朱利亚集）可以是一个蜘蛛网。

超越对应：当富克斯群接管整函数的盆域时

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。

超越整映射Sierpinski曲线朱利亚集的非着陆毛

Non-landing hairs in Sierpinski curve Julia sets of transcendental entire maps

本文考虑由$f_a(z)=a(z-(1-a))\exp(z+a)$给出的超越整映射族，其中$a$是复参数.每个映射在$z=-a$有一个超吸引不动点，在$z=0$有一个渐近值。对于$a>1$，已知$f_a$的Julia集同胚于Sierpienski通用曲线，因此包含任何一维平面连续体的嵌入副本。在本文中，我们研究的Julia集，可以定义在一个组合的方式下的子连续统。特别是，我们表明存在的非着陆头发与规定的组合嵌入在朱莉娅集的所有参数$a\geq 3$。我们还研究了非着陆毛与$z=-a$的直接吸引区域之间的关系。即使每一个非着陆的头发在一个点上积累到直接盆地的边界上，它的闭包仍然成为Julia集的不可分解的子连续统。

两个可交换超越整函数的动力学集合相等

Equality of the dynamical sets of two commuting transcendental entire functions

本文研究了两个可交换的超越整函数f和g的动力学性质，其中g的形式为af^p + b，其中a,b∈C，p∈N，且a≠0,1。研究发现f和g的逃逸集、填充Julia集和蹦床集都相同。作为直接结果，我们得出f和g的Julia集相同。该定理扩展了Poon和Yang1998年的结果。此外，遵循Wang和Yang的研究，我们考虑了一个非常数多项式Q和可交换的整个函数f和g，满足关系Q(g)=aQ(f)+b，其中a≠0,1，b∈C。在更一般的情况下，我们证明了f和g的逃逸集、填充Julia集和蹦床集相等。

超越半群的逃逸集与朱利亚集

Escaping set and Julia set of transcendental semigroups

利用Fatou-Julia理论讨论了超越整函数半群的动力学，给出了超越半群的逃逸集和朱利亚集完全不变性的条件.关于极限函数和后奇异集的结果也得到了。此外，还给出了双曲和后奇异有界超越半群类。给出了超越半群游荡域不存在的判定准则。

超越函数的无限熵等分布测度

Equidistribution Measures of infinite entropy for Transcendental Functions

本文为超越整函数构造了无限熵不变测度，通过嵌入符号动力系统和传递算子两种方法，证明原像及周期点等分布，并指出测度非唯一。

整超越映射的一个分离定理

A separation theorem for entire transcendental maps

本文研究了一类映射的周期点的分布，即具有有界奇异值集的有穷级整超越函数或其组合。修复$p\in\N$，并假设所有在$f^p$下不变的动态光线都是陆地。内部$p$ -周期点是$f^p$的不动点，它不是在$f^p$下不变量的任何周期射线的着陆点。属于吸引、西格尔或克里默循环的点是内部周期点的例子。对于上述功能，我们表明，射线是不变的下$f^p$，连同他们的着陆点，分离的平面成100多个区域，每一个恰好包含一个内部$p-$周期点或一个抛物线立即盆不变下$f^p$。这个结果推广了多项式的Goldberg-Milnor分离定理，并有几个推论。例如，两个周期性Fatou分量总是可以被一对同时入射的周期性射线分开; Siegel圆盘的边界上不可能有Cremer点;有界Siegel圆盘的“隐藏分量”要么是游荡域，要么是Siegel圆盘本身的预周期;或者，无论奇异值的数量如何，在任何给定的周期中，只有100多个非排斥循环。

Dynamics of Nearly Abelian Transcendental Semigroup

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.

超越整函数复合的后奇异集的结果

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。

Transcendental Julia sets of positive finite Lebesgue measure

We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.

几何有限的超越整函数

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。

在S类中没有含实乘子的整函数

No entire function with real multipliers in class S

我们证明了在S类中不存在具有所有排斥周期轨道的实乘子的整超越函数。

整超越函数上的形变

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。

实超越整函数类和圆映射类的双曲性稠密性

Density of hyperbolicity for classes of real transcendental entire functions and circle maps

我们证明了密度的双曲空间（一）真正的超越整函数，有界的真正的线，其奇异集是有限的和真正的和（ii）超越自我映射的穿孔平面保持圆，其奇异集（除了零和无穷大）包含在圆。特别是，我们证明了密度的双曲在著名的Arnol'd家庭的圆地图及其推广，并解决了一些其他开放的问题，这些功能，包括三个apturtures由德梅洛，Salomão和巴尔加斯。我们还证明了密度的（真正的）双曲的某些家庭（i），但没有有界性条件。我们的结果适用，特别是，当所讨论的功能只有100多个临界点和渐近奇点，或当没有渐近值和程度的临界点是一致有界的。

共轭整函数的动力学

On dynamics of semiconjugated entire functions

设$g$和$h$是超越整函数，且$f$是复平面到自身的连续映射，其中有$f\circ g=h\circ f.$，则称$g$和$h$被$f$共轭，称$f$为共轭。研究了双共轭超越整函数$g$和$h$的动力学性质，给出了双共轭$f$将一个整函数的法图集带入双共轭中出现的另一个整函数的法图集的几个条件.证明了在一定条件下，在共轭中出现的整函数的增长性，其复合导数的渐近值的集合是有界的。

超越动力学中的逃逸集

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。

Infinite-dimensional Thurston theory and transcendental dynamics IV: dependence on parameters and escape on (pre-)periodic rays

We consider transcendental entire functions that are compositions of a polynomial and the exponential for which all singular values escape on disjoint rays. Based on their classification in [B3] we investigate their dependence on parameters, that is, on the potentials and external addresses of singular values. Using continuity argument we generalize the classification in [B3] to the case when dynamic rays containing singular values are (pre-)periodic.

A generalized family of transcendental functions with one dimensional Julia sets

A generalized family of transcendental (non-polynomial entire) functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to one. Further, there exist multiply connected wandering domains, the dynamics can be completed described, and for any $s\in(0,+\infty]$, there is a function taken from this family with the order of growth $s$. Baker proved that the Hausdorff dimension of the transcendental function is no less than one in 1975, the minimum value was obtained via an elegant construction by Bishop in 2018. The order of growth is zero in Bish…

关于可交换超越整函数及其动力学的综述

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。

Stable components in the parameter plane of transcendental functions of finite type

We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call {\em shell components}, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components…

超越整函数逃逸动力学的刚性

Rigidity of escaping dynamics for transcendental entire functions

本文证明了B类中超越整函数的Boettcher定理的一个类似定理。更准确地说，设f和g是具有有界奇异值集的整函数，并假设f和g属于相同的参数空间（即，在Eremenko和Lyubich的意义上是 * 拟共形等价的 *）。然后f和g是共轭时，限制到一组点仍然在一些足够小的邻域无穷迭代。此外，这种共轭性推广到平面的拟共形自映射。我们还证明了这种共轭性本质上是唯一的。特别是，我们证明了Eremenko-Lyubich类函数f在其逃逸集上没有不变的线场。最后证明了属于同一参数空间的任意两个双曲Eremenko-Lyubich类函数f和g在其逃逸点集上共轭。

Iterated Monodromy Groups of Entire Maps and Dendroid Automata

This paper discusses iterated monodromy groups for transcendental functions. We show that for every post-singularly finite entire transcendental function, the iterated monodromy action can be described by bounded activity automata of a special form, called "dendroid automata". In particular, we conclude that the iterated monodromy group of a post-singularly finite entire function is amenable if and only if the monodromy group is.

Splitting hairs with transcendental entire functions

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…

梳理超越整函数的毛发

Brushing the hairs of transcendental entire functions

设f是Eremenko-Lyubich类中的有穷级双曲超越整函数（或这类映射的有限复合），且f有唯一的Fatou分支.我们表明，朱利亚集的$f$是康托花束，即环境同胚的直刷的意义上的Aarts和Oversteegen。特别地，我们证明了任意两个这样的朱利亚集是环境同胚的.我们还表明，如果$f\in\B$有有穷级（或是一个有限的组成，这样的地图），但不一定是双曲的，那么朱利亚集的f包含一个康托花束。作为我们的证明的一部分，我们描述，为任意函数$f\in\B$，一个自然的紧化的动态平面通过添加一个“圆的地址”在无穷大。

超越函数随机动力学的实解析性

Real Analyticity for random dynamics of transcendental functions

建立了超越函数随机动力学中期望压强和不变密度的解析结果。这些都是由一个完善的工作由鲁导致一个简单的方法来分析。我们在非常温和的动力学假设下工作。仅假设Perron-Frobenius算子的迭代收敛。我们还提供了一个Bowen的公式表示的几乎必然的豪斯多夫维数的径向纤维朱利亚集的零的期望压力函数。我们的主要应用状态的变化，这方面的适当的双曲型随机系统的整个或亚纯函数的真正的解析性。

与超越整函数提升相关的内函数

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。

A new family of entire functions with no wandering domains

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.

无无界Fatou分支的小增长函数

Functions of small growth with no unbounded Fatou components

证明了$\cos πρ$定理的一种形式，该定理给出了零阶超越整函数的最小模的强估计。我们还证明了Hinkkanen的一个结果的推广，给出了一个充分条件的超越整函数没有无界Fatou组件。这两个结果使我们能够证明，有一大类零阶整函数没有无界Fatou分量。另一方面，我们给出的例子表明，实际上有功能的零阶不仅不满足Hinkkanen的条件，但也不满足我们更一般的条件。我们还给出了一个新的正则性条件，该条件足以保证一个小于1/2阶的超越整函数没有无界的Fatou分量。最后，我们观察到，这里给出的所有条件，保证了一个超越整函数没有无界Fatou组件，也保证了逃逸集是连接的，从而回答了一个问题，Eremenko为这样的功能。

Thermodynamic formalism and integral means spectrum of asymptotic tracts for transcendental entire functions

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…

拟正则映射的快速逃逸集

The fast escaping set for quasiregular mappings

一个超越整函数的快速逃逸集是所有在迭代下趋于无穷大的点的集合，其速度与函数的增长速度一样快。我们研究了类似的一套拟正则映射在更高的维度和显示，除其他事项外，各种等价定义的快速逃逸集的超越整函数在平面上也符合拟正则映射。我们还展示了一类拟正则映射的快速逃逸集具有蜘蛛网的结构。

关于吸引盆地边界的维数

On the dimension of the boundaries of attracting basins of entire maps

本文研究了超越整个映射吸引盆地边界的维数，证明了其超几何维数大于1，并展示了其边界不可能是光滑或可测曲线。

Transcendental Julia Sets with Fractional Packing Dimension

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.

超越函数的随机动力学

Random Dynamics of Transcendental Functions

本文研究了有穷级双曲整函数和亚纯函数的随机动力学，其导数在无穷远处满足一定的增长条件。这个类包含了大多数经典的超越函数族，并且远远超出了它。基于Nevanlinna的值分布理论的统一版本，我们首先建立了一个数学形式主义，特别是，产生独特的几何和纤维不变的吉布斯态。此外，谱间隙性质的相关性的指数衰减和中心极限定理的相关转移算子。这一部分依赖于我们的建设新的积极不变锥，适应于无界相空间的设置。这种设置排除了希尔伯特度量和通常的压缩原理的使用。然而，这些锥体使我们能够应用源于Bowen最初方法的收缩论点。

Combinatorics of criniferous entire maps with escaping critical values

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…