Markus Förster, Lasse Rempe, Dierk Schleicher
We give a complete classification of the set of parameters $κ$ for which the singular value of $E_κ:z\mapsto \exp(z)+κ$ escapes to infinity under iteration. In particular, we show that every path-connected component of this set is a curve to infinity.
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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 different rates of escape of points under iteration by holomorphic self-maps of $\mathbb C^*=\mathbb C\setminus\{ 0\}$ for which both 0 and $\infty$ are essential singularities. Using annular covering lemmas we construct different types of orbits, including fast escaping and arbitrarily slowly escaping orbits to either 0, $\infty$ or both. We also prove several properties about the set of fast escaping points for this class of functions. In particular, we show that there is an uncountable collection of disjoint sets of fast escaping points each of which has the Julia set as its bo…
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 develop an abstract model for the dynamics of an exponential map $z\mapsto \exp(z)+κ$ on its set of escaping points and, as an analog of Boettcher's theorem for polynomials, show that every exponential map is conjugate, on a suitable subset of its set of escaping points, to a restriction of this model dynamics. Furthermore, we show that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an analog of Douady's "pinched disk model" for the Julia sets of these maps. On the other hand, we show that two exponential maps are gene…
Dynamical (non-)recurrence of escaping exponential maps
该研究针对奇异值缓慢逃逸至无穷的指数映射，构造了回复与非回复的例子，补充了相关学者的早期结果，还给出了奇异值逃逸速率任意慢的参数化例子。
We show that for many complex parameters a, the set of points that converge to infinity under iteration of the exponential map f(z)=e^z+a is connected. This includes all parameters for which the singular value escapes to infinity under iteration.
We extend results about the dimension of the radial Julia set of certain exponential functions to quasiregular Zorich maps in higher dimensions. Our results improve on previous estimates of the dimension also in the special case of exponential functions.
Relation between Hénon maps with biholomorphic escaping sets
该研究证明两个具有双全纯等价逃逸集的 Hénon 映射可通过 $\mathbb{C}^2$ 中的两个仿射自同构复合建立仿射共轭关系。
We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity `slowly', and which have Hausdorff dimension equal to 1. We prove these results by using the idea of an annular itinerary. In the case of a general transcendental entire function we show that one of these sets, the uniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.
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…
Ergodic exponential maps with escaping singular behaviours
研究构造了遍历的指数映射，其奇点值在迭代中趋于无穷，与Lyubich 1987年结果形成对比。
We show that the points that converge to infinity under iteration of the exponential map form a connected subset of the complex plane.
Escaping Sets of Skew Products of Hénon maps
本研究将Hénon映射逃逸集的描述推广至斜积，构造覆盖空间探究其全局逃逸集的解析结构与双全纯等价性，还研究了以非紧复平面为底的斜积的逃逸区域。
Rigidity of the escaping set of certain Hénon maps
该研究证明了特定Hénon映射的逃逸集在复平面自同构作用下的刚性，并展示了其自同构群的结构。
It is shown that for quasiregular maps of positive lower order the Julia set coincides with the boundary of the fast escaping set.
Let $E_{\la}(z)=\la {\rm exp}(z), \ λ\in \mathbb C$ be the complex exponential family. For all functions in the family there is a unique asymptotic value at 0 (and no critical values). For a fixed $\la$, the set of points in $\mathbb C$ with orbit tending to infinity is called the escaping set. We prove that the escaping set of $E_{\la}$ with $\la$ Misiurewicz (that is, a parameter for which the orbit of the singular value is strictly preperiodic) is a connected set.
We study the escaping set of functions in the class $\mathcal B^*$, that is, holomorphic functions $f:\mathbb C^*\to\mathbb C^*$ for which both zero and infinity are essential singularities, and the set of singular values of $f$ is contained in a compact annulus of $\mathbb C^*$. For functions in the class $\mathcal B^*$, escaping points lie in their Julia set. If $f$ is a composition of finite order transcendental self-maps of $\mathbb C^*$ (and hence, in the class $\mathcal B^*$), then we show that every escaping point of $f$ can be connected to one of the essential singularities by a curve …
We study the dynamics of a collection of families of transcendental entire functions which generalises the well-known exponential and cosine families. We show that for functions in many of these families the Julia set, the escaping set and the fast escaping set are all spiders' webs of positive area. This result is unusual in that most of these functions lie outside the Eremenko-Lyubich class B. This is also the first result on the area of a spider's web.
On automorphisms of super-level sets of Green's functions of Hénon maps
本文研究Hénon映射逃逸集及相关格林函数超水平集的自同构，得到逃逸集自同构的局部标准型，刻画了保持逃逸集的$\text{C}^2$整体自同构，还描述了格林函数超水平集的自同构。
It is known that, for many transcendental entire functions in the Eremenko-Lyubich class $\mathcal{B}$, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are criniferous. In this paper, we extend this result to a new class of maps in $\mathcal{B}$. Furthermore, we show that if a map belongs to this class, then its Julia set contains a Cantor bouquet; in other words, it is a subset of $\mathbb{C}$ ambiently homeomorphic to a straight brush.
We consider the case of an exponential map for which the singular value is accessible from the set of escaping points. We show that there are dynamic rays of which do not land. In particular, there is no analog of Douady's ``pinched disk model'' for exponential maps whose singular value belongs to the Julia set. We also prove that the boundary of a Siegel disk $U$ for which the singular value is accessible both from the set of escaping points and from $U$ contains uncountably many indecomposable continua.
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 wish to investigate some elementary problems concerning topological dynamics revolving around our proposed definition of escaping set. We also discuss the notion of escaping set in the induced dynamics of the hyperspace. Moreover, we extend the notion of limit set and escaping set for the general semigroup generated by continuous self maps.
An analogue of Green's Functions for Quasiregular Maps
本文为二次次数且常数复伸缩的平面拟正则映射构造格林函数类比，其半共轭到实数平方映射，但正则性不同于多项式情形，用于研究逃逸集边界并展示新行为。
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…
In 1988, Mayer proved the remarkable fact that infinity is an explosion point for the set of endpoints of the Julia set of an exponential map that has an attracting fixed point. That is, the set is totally separated (in particular, it does not have any nontrivial connected subsets), but its union with the point at infinity is connected. Answering a question of Schleicher, we extend this result to the set of "escaping endpoints" in the sense of Schleicher and Zimmer, for any exponential map for which the singular value belongs to an attracting or parabolic basin, has a finite orbit, or escapes …
We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential $\exp(z)+a$ when $a\in (-\infty,-1)$. We show neither space is homeomorphic to the whole set of endpoints. This follows from a general result stating that for every transcendental entire function $f$, the escaping Julia set $I(f)\cap J(f)$ is first category.
We consider the dynamics of transcendental self-maps of the punctured plane, $\mathbb{C}^*=\mathbb{C}\setminus \{0\}$. We prove that the escaping set $I(f)$ is either connected, or has infinitely many components. We also show that $I(f)\cup \{0,\infty\}$ is either connected, or has exactly two components, one containing $0$ and the other $\infty$. This gives a trichotomy regarding the connectivity of the sets $I(f)$ and $I(f)\cup \{0,\infty\}$, and we give examples of functions for which each case arises. Finally, whereas Baker domains of transcendental entire functions are simply connected, w…
For many transcendental entire functions, the escaping set has the structure of a Cantor bouquet, consisting of uncountably many disjoint curves. Rippon and Stallard showed that there are many functions for which the escaping set has a new connected structure known as an infinite spider's web. We investigate a connection between these two topological structures for a certain class of sums of exponentials.
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…
Let $f$ be a transcendental entire function and let $U$ be a univalent Baker domain of $f$. We prove a new result about the boundary behaviour of conformal maps and use this to show that the non-escaping boundary points of $U$ form a set of harmonic measure zero with respect to $U$. This leads to a new sufficient condition for the escaping set of $f$ to be connected, and also a new general result on Eremenko's conjecture.
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…
The study of escape rates for a ball in a dynamical systems has been much studied. Understanding the asymptotic behavior of the escape rate as the radius of the ball tends to zero is an especially subtle problem. In the case of hyperbolic conformal systems this has been addressed by various authors. In this paper we consider a far more general realm of conformal maps where the analysis is correspondingly more complicated. We prove the existence of escape rates and calculate them in the context of countable alphabets, either finite or infinite, uniformly contracting conformal graph directed Mar…
Entire maps with rational preperiodic points and multipliers
本文证明了对于非实数域的数域，存在预周期点和乘数均位于其中的整映射稠密集，并发现了一个乘数全为有理数且不共轭于仿射映射的逃逸类二次映射。
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-…
Rigidity of the escaping set of polynomial automorphisms of $\mathbb{C}^2$
研究多项式 $\mathbb{C}^2$ 自动同构的逃逸集刚性，证明其自同构群为有限循环群。
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 consider the Hausdorff measure of Julia sets and escaping sets of exponential maps with respect to certain gauge functions. We give conditions on the growth of the gauge function which imply that the measure is zero or infinity, respectively.
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。
We consider the family of holomorphic maps e^z+c and show that fibers of postcritically finite parameters are trivial. This can be seen as the first and simplest class of non-escaping parameters for which we can obtain results about triviality of fibers in the exponential family.
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。
We give a complete combinatorial description of the bifurcation structure in the space of exponential maps $z\mapsto\exp(z)+κ$. This combinatorial structure is the basis for a number of important results about exponential parameter space. These include the fact that every hyperbolic component has connected boundary, a classification of escaping parameters, and the fact that all dynamic and parameter rays at periodic addresses land.
We prove the exponential decay of correlations for C^α-observables (0<α=<2) for generic birational maps of P^k à la Bedford-Diller. In the particular case of regular birational maps, we give a better estimate of the speed of the decay, getting results as sharp as Dinh's results for Hénon maps.
The Julia set of the exponential family $E_κ:z\mapstoκe^z$, $κ>0$ was shown to be the entire complex plane when $κ>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<κ\leq1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the entire $\mathbb{R}^3$ generalizing Misiurewicz's result. Moreover, we show that the periodic points…
Let $f$ be a Hénon-Sibony map (regular polynomial automorphism) of $\mathbb{C}^k$ and let $μ$ be the equilibrium measure of $f$. In this paper we prove that $μ$ is exponentially mixing for plurisubharmonic test functions.
In this paper we show an infinite measure set of exponentially escaping orbits for a resonant Fermi accelerator, which is realised as a square billiard with a periodically oscillating platform. We use normal forms to describe how the energy changes in a period and we employ techniques for hyperbolic systems with singularities to show the exponential drift of these normal forms on a divided time-energy phase.
We show that an exponential map $f_c(z)=e^z+c$ whose singular value $c$ is combinatorially non-recurrent and non-escaping is uniquely determined by its combinatorics, i.e. the pattern in which its dynamic rays land together. We do this by constructing puzzles and parapuzzles in the exponential family. We also prove a theorem about hyperbolicity of the postsingular set in the case that the singular value is non-recurrent. Finally, we show that boundedness of the postsingular set implies combinatorial non-recurrence if $c$ is in the Julia set.
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…
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.
In [arXiv:1904.12333], we introduced the concept of escaping set in general setting for a topological space and extended the notion of limit set and escaping set for the general semigroup generated by continuous self maps. In this paper we continue with extending the other notions of recurrence for the generalized semigroup analogs to their counterpart in the classical theory of dynamics. We discuss the concept of periodic point, nonwandering point and chain recurrent point in the more general setting of a semigroup and establish the correlation between them.
The set of points that escape to infinity under iteration of a cosine map, that is, of the form $C_{a,b} \colon z \mapsto ae^z+be^{-z}$ for $a,b\in \mathbb{C}^\ast$, consists of a collection of injective curves, called dynamic rays. If a critical value of $C_{a,b}$ escapes to infinity, then some of its dynamic rays overlap pairwise and \textit{split} at critical points. We consider a large subclass of cosine maps with escaping critical values, including the map $z\mapsto \cosh(z)$. We provide an explicit topological model for their dynamics on their Julia sets. We do so by first providing a mo…
We consider parameters $λ$ for which $0$ is preperiodic under the map $z\mapstoλe^z$. Given $k$ and $l$, let $n(r)$ be the number of $λ$ satisfying $0<|λ|\leq r$ such that $0$ is mapped after $k$ iterations to a periodic point of period $l$. We determine the asymptotic behavior of $n(r)$ as $r$ tends to $\infty$.
Exponential mixing of all orders and CLT for generic birational maps of $\mathbb{P}^k$
针对$\boldsymbol{\text{mathbb{P}}^k}$上一般双有理映射的最大熵测度，推广了Hénon映射下的所有阶指数混合及Hölder可观测量CLT结论，通过引入合适的检验函数空间解决了不定集导致的Hölder映射迭代不稳定问题。
We answer a question of Schleicher by showing that, for an exponential map with nonescaping singular value, every periodic ray lands. This is an analog of a theorem of Douady and Hubbard concerning polynomials. We also prove a partial converse: there are periodic external rays landing at all periodic points, with the exception of at most one periodic orbit.
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。
We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard's celebrated theorem that (the boundary of) the Mandelbrot set is connected. For these parameter spaces, a fundamental conjecture is that hyperbolic dynamics is dense. For quadratic polynomials, this would follow from the famous stronger conjecture that the bifurcation locus (or equivalently the Mandelbrot set) is locally connected. It turns out that a formally slightly weaker statement is …
Spiders' webs in the Eremenko-Lyubich class
本文证明整函数 cosh(z) 的逃逸集乃至快速逃逸集均呈“蜘蛛网”结构，反驳并解答了相关猜想与问题。
Non-analyticity of hairs of exponential maps
本文证明了指数映射 $f_{\lambda}(z)=\lambda e^z$（$0<\lambda<1/e$）的 Julia 集中的毛发除平凡情形外均非解析，解决了长期未决问题。
Dimension of the accumulation set of any hair for the exponential map
该研究针对复平面上指数映射，证明指数有界行程对应的共享点集Λ_c的豪斯多夫维数为1，而部分行程的Λ_c具有不可分解连续统等复杂拓扑结构。
Since 1984, many authors have studied the dynamics of maps of the form $\mathcal{E}_a(z) = e^z - a$, with $a > 1$. It is now well-known that the Julia set of such a map has an intricate topological structure known as a Cantor bouquet, and much is known about the dynamical properties of these functions. In recent papers some of these ideas have been generalised to a class of quasiregular maps in $\mathbb{R}^3$, which, in a precise sense, is analogous to the class of maps of the form $\mathcal{E}_a$. Our goal in this paper is to make similar generalisations in $\mathbb{R}^2$. In particular, we s…
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