Walter Bergweiler, Bogusława Karpińska, Gwyneth M. Stallard
Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.
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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 prove the existence of rational maps whose Julia sets are Sierpiński carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely renormalizable. We also construct some Sierpiński carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number $s\in(1,2)$, we prove the existence of Sierpiński carpet Julia sets having Hausdorff dimension exactly $s$.
We show that for each $d\in (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.
We prove the existence of Cantor Julia sets with Hausdorff dimension two. In particular, such examples can be found in cubic polynomials. The proof is based on the characterization of the parameter spaces and dynamical planes of cubic polynomials by Branner-Hubbard, and the parabolic bifurcation theory developed by Shishikura in 1990s.
We give examples of infinitely renormalizable quadratic polynomials $F_c: z\maps to z^2+c$ with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrar y close to 1. The combinatorics of the renormalization involved is close to the Chebyshev one . The argument is based upon a new tool, a ``Recursive Quadratic Estimate'' for the Poincaré series of an infinitely re normalizable map.
For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which…
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。
Devaney and Krych showed that for $0<λ<1/e$ the Julia set of $λe^z$ consists of pairwise disjoint curves, called hairs, which connect finite points, called the endpoints of the hairs, with $\infty$. McMullen showed that the Julia set has Hausdorff dimension $2$ and Karpińska showed that the set of hairs without endpoints has Hausdorff dimension $1$. We study for which gauge functions the Hausdorff measure of the set of hairs without endpoints is finite.
Sun Daochun and Yang Lo have shown that many results of the Fatou-Julia iteration theory of rational functions extend to quasiregular self-maps of the Riemann sphere for which the degree exceeds the dilatation. We show that in this context, in contrast to the case of rational functions, the Julia set may have Hausdorff dimension zero. On the other hand, we exhibit a gauge function depending on the degree and the dilatation such that the Hausdorff measure with respect to this gauge function is always positive, but may be finite.
We survey the definition of the radial Julia set of a meromorphic function (in fact, more generally, any "Ahlfors islands map"), and give a simple proof that the Hausdorff dimension of the reduced Julia set always coincides with the hyperbolic dimension.
We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman-Valiron theory.
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.
In this paper, we study the large scaled geometric structure of Julia sets of entire and meromorphic functions. Roughly speaking, the structure gives us some asymptotic information about the Julia set near the essential singularity. We will show that part of this structure are determined by the transcendental directions coming from function theoretic point of view.
For any $ δ>0$ we construct an entire function $f$ with three singular values whose Julia set has Hausdorff dimension at most $1=δ$. Stallard proved that the dimension must be strictly larger than 1 whenever $f$ has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known.
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.
We show that for large classes of entire functions the Julia set and the escaping set have packing dimension two. For example, this is the case for entire functions which are bounded on a curve tending to infinity. More generally, we show that the result holds under suitable growth conditions involving the minimum and maximum modulus.
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.
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.
It is well-known that the Julia set J(f) of a rational map is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this article we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f in R^n is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.
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 consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that …
In this paper we study the dynamics of regular polynomial automorphisms of C^n. These maps provide a natural generalization of complex Henon maps in C^2 to higher dimensions. For a given regular polynomial automorphism f we construct a filtration in C^n which has particular escape properties for the orbits of f. In the case when f is hyperbolic we obtain a complete description of its orbits. In the second part of the paper we study the Hausdorff and box dimension of the Julia sets of f. We show that the Julia set J has positive box dimension, and (provided f is not volume preserving) that the …
Hausdorff dimension in quasiregular dynamics
本文研究三维空间中拟正则自映射，证明其快逃逸集的Hausdorff维数可取[1,3]内任意值，并在增长条件下估计Julia集的维数。
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.
We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.
On hyperbolic dimension gap for entire functions
该研究解决了双曲维数小于Julia集Hausdorff维数的整函数是否仅存在于后者维数为2的情况的问题，构造出了Julia集Hausdorff维数小于2且双曲维数仍小于该值的双曲整函数，填补了相关维数间隙研究的空白。
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…
We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen's formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition. What we discuss is the `backward iterated function systems', and since the generator maps are not injective in general, we need some efforts to estimate the measure of disks.
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 study the family of renormalization transformations of the generalized $d$--dimensional diamond hierarchical Potts model in statistical mechanic and prove that their Julia sets and non-escaping loci are always connected, where $d\geq 2$. In particular, we prove that their Julia sets can never be a Sierpiński carpet if the parameter is real. We show that the Julia set is a quasicircle if and only if the parameter lies in the unbounded capture domain of these models. Moreover, the asymptotic formula of the Hausdorff dimension of the Julia set is calculated as the parameter tends to infinity.
Recently Merenkov and Sabitova introduced the notion of a homogeneous planar set. Using this notion they proved a result for Sierpi${ń}$ski carpet Julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles to the Hausdorff dimension of the Julia set. We extend this theorem to Julia sets (not necessarily Sierpi${ń}$ski carpets) of semi-hyperbolic rational maps, and prove a stronger version of the theorem that was conjectured by Merenkov and Sabitova.
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.
The Hausdorff dimension of escaping sets of meromorphic functions in the Speiser class
本研究证明Bergweiler和Kotus给出的Eremenko-Lyubich类半纯函数逃逸集Hausdorff维数上界在Speiser类中也精确，还构造了具指定Julia集与逃逸集维数的Speiser类半纯函数。
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…
Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture
本文研究了分离型整个函数的Julia集拓扑性质，证明了其Julia连续体为弧状连续体，并构造了具有特定性质的函数，与Eremenko猜想相关。
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。
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。
In 1984 Devaney and Krych showed that for the exponential family $λe^z$, where $0<λ<1/e$, the Julia set consists of uncountably many pairwise disjoint simple curves tending to $\infty$, which they called hairs. Viana proved that these hairs are smooth. Barański as well as Rottenfusser, Rückert, Rempe and Schleicher gave analogues of the result of Devaney and Krych for more general classes of functions. In contrast to Viana's result we construct in this article an entire function, where the Julia set consists of hairs, which are nowhere differentiable.
This paper deals with both complex dynamical systems and conformal iterated function systems. We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a $d$-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2, then typ…
The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.
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 article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given one-dimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This extends and unifies classical results due to Baker, Beardon, Eremenko, Levin, Przytycki and others. The proof involves entire curves and positive currents.
We present a simple proof of Tan's theorem on asymptotic similarity between the Mandelbrot set and Julia sets at Misiurewicz parameters. Then we give a new perspective on this phenomenon in terms of Zalcman functions, that is, entire functions generated by applying Zalcman's lemma to complex dynamics. We also show asymptotic similarity between the tricorn and Julia sets at Misiurewicz parameters, which is an antiholomorphic counterpart of Tan's theorem.
Let $f(z) = z^2 + c$ be a quadratic polynomial, with c in the Mandelbrot set. Assume further that both fixed points of f are repelling, and that f is not renormalizable. Then we prove that the Julia set J of f is holomorphically removable in the sense that every homeomorphism of the complex plane to itself that is conformal off of J is in fact conformal on the entire complex plane. As a corollary, we deduce that the Mandelbrot Set is locally connected at such c.
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.
We show, in an elementary way, that the Julia set of one-complex-variable entire functions is nonempty and perfect.
For 0 < c < 1/e the Julia set of f(z) = c exp(z) is an uncountable union of pairwise disjoint simple curves tending to infinity [Devaney and Krych 1984], the Hausdorff dimension of this set is two [McMullen 1987], but the set of curves without endpoints has Hausdorff dimension one [Karpinska 1999]. We show that these results have three-dimensional analogues when the exponential function is replaced by a quasiregular self-map of three-space introduced by Zorich.
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 show that the iterative logarithm of each non-linear entire function is differentially transcendental over the ring of entire functions, and we give a sufficient criterion for such an iterative logarithm to be differentially transcendental over the ring of convergent power series.
We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed by Lindsey and Younsi and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show that taking certain classes of nodes with subexponential growth of Lebesgue constants improves the approximation rate. To this end we investigate properties of some arrays of points in $\mathbb{C}$. In particular we prove subexponential growth of Lebesgue constants for pseudo Leja sequences with bounded Edrei growth on finite unions of quasiconfo…
Horn maps of semi-parabolic Hénon maps
本文证明了半抛物Hénon映射的Horn映射满足弱Ahlfors岛性质，从而统一了Julia集的两种定义，并证明小扰动可使前向Julia集的Hausdorff维数任意接近4。
We show that there exists a hyperbolic entire function of finite order of growth such that the hyperbolic dimension---that is, the Hausdorff dimension of the set of points in the Julia set of whose orbit is bounded---is equal to two. This is in contrast to the rational case, where the Julia set of a hyperbolic map must have Hausdorff dimension less than two, and to the case of all known explicit hyperbolic entire functions. In order to obtain this example, we prove a general result on constructing entire functions in the Eremenko-Lyubich class with prescribed behavior near infinity, using Cauc…
The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2$. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2.$ We prove that if the Julia set of a semigroup $G$ which is generated by endomorphisms of maximal generic rank $k$ in $\mathbb{C}^k$ contains an isolated point, then $G$ must contain an element that is conjugate to an upper triangular automorphism of $\mathbb{C}^k.$ This generalizes a t…
We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup $G$ generated by a compact family of rational maps on the Riemann sphere of degree two or greater, the Julia set of any subsemigroup of $G$ is uniformly perfect. We define the semi-hyperbolicity of dynamics on fiber bundles and show that, if the dynamics on a fiber bundle is sem…
Simple Proofs for the Derivative Estimates of the Holomorphic Motion near Two Boundary Points of the Mandelbrot Set
本文针对复二次族朱利亚集在曼德博集合边界点1/4和-2附近的全纯运动，给出导数的一致渐近估计，并由此证明朱利亚集的豪斯多夫收敛及精确距离公式。
In this paper we mainly investigate the radial distribution of Julia set of derivatives of entire solutions of some complex linear differential equations. Under certain conditions, we find the lower bound of it which improve some recent results.
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 consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the function of the Hausdorff dimension of the Julia set is real-analytic and plurisubharmonic. Moreover, we provide an extensive collection of classes of examples of analytic fam…
We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected (\cite{Su01,Su9}). In this paper, we show that for any two distinct Fatou components of ce…
Julia directions for holomorphic curves
该研究证明了映入复射影簇的整全纯映射的Picard型定理，其具备局部特性可证Julia方向存在，同时举例说明相关Borel定理无局部对应结论，推进了复分析领域相关研究。
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