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Accessible Boundary Points in the Shift Locus of a Family of Meromorphic Functions with Two Finite Asymptotic Values 具有两个有限渐近值的亚纯函数族移位轨迹上的可及边界点
Q3 Mathematics Pub Date : 2021-01-07 DOI: 10.1007/s40598-020-00169-1
Tao Chen, Yunping Jiang, Linda Keen

In this paper, we continue the study, began in Chen et al. (Slices of parameter space for meromorphic maps with two asymptotic values, arXiv:1908.06028, 2019), of the bifurcation locus of a family of meromorphic functions with two asymptotic values, no critical values, and an attracting fixed point. If we fix the multiplier of the fixed point, either of the two asymptotic values determines a one-dimensional parameter slice for this family. We proved that the bifurcation locus divides this parameter slice into three regions, two of them analogous to the Mandelbrot set and one, the shift locus, analogous to the complement of the Mandelbrot set. In Fagella and Keen (Stable components in the parameter plane of meromorphic functions of finite type, arXiv:1702.06563, 2017) and Chen and Keen (Discrete and Continuous Dynamical Systems 39(10):5659–5681, 2019), it was proved that the points in the bifurcation locus corresponding to functions with a parabolic cycle, or those for which some iterate of one of the asymptotic values lands on a pole are accessible boundary points of the hyperbolic components of the Mandelbrot-like sets. Here, we prove these points, as well as the points where some iterate of the asymptotic value lands on a repelling periodic cycle are also accessible from the shift locus.

在本文中,我们继续从Chen等人开始的研究。(具有两个渐近值的亚纯映射的参数空间切片,arXiv:1908.060282019),关于具有两个渐进值、无临界值和一个吸引不动点的亚纯函数族的分支轨迹。如果我们固定不动点的乘数,那么两个渐近值中的任何一个都会确定该族的一维参数切片。我们证明了分叉轨迹将这个参数切片划分为三个区域,其中两个区域类似于Mandelbrot集,另一个区域是移位轨迹,类似于Mandel brot集的补集。在Fagella和Keen(有限型亚纯函数参数平面上的稳定分量,arXiv:1702.065632017)和Chen和Keen,或者其中一个渐近值的一些迭代落在极点上的那些是Mandelbrot样集的双曲分量的可访问边界点。在这里,我们证明了这些点,以及渐近值的一些迭代落在排斥周期周期上的点,也可以从移位轨迹访问。
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引用次数: 6
Simplicity of Spectra for Bethe Subalgebras in ({mathrm {Y}}({mathfrak {gl}}_2)) {mathrm{Y}}({mashfrak{gl}}2)中Bethe子代数的谱的简单性
Q3 Mathematics Pub Date : 2021-01-07 DOI: 10.1007/s40598-020-00171-7
Inna Mashanova-Golikova

We consider Bethe subalgebras B(C) in the Yangian ({mathrm {Y}}({mathfrak {gl}}_2)) with C regular (2times 2) matrix. We study the action of Bethe subalgebras of ({mathrm {Y}}({mathfrak {gl}}_2)) on finite-dimensional representations of ({mathrm {Y}}({mathfrak {gl}}_2)). We prove that B(C) with real diagonal C has simple spectrum on any irreducible ({mathrm {Y}}({mathfrak {gl}}_2))-module corresponding to a disjoint union of real strings. We extend this result to limits of Bethe algebras. Our main tool is the computation of Shapovalov-type determinant for the nilpotent degeneration of B(C).

我们考虑Yangian({mathrm{Y}}({/mathfrak{gl}}2))中具有C正则(2×2)矩阵的Bethe子代数B(C)。我们研究了({mathrm{Y}}({ mathfrak{gl}}2))的Bethe子代数对(}mathrm{Y}}({athfrak{gl}}_2))有限维表示的作用。我们证明了实对角线为C的B(C)在对应于实串的不相交并集的任何不可约({mathrm{Y}})({ mathfrak{gl}2))-模上具有简单谱。我们将这个结果推广到Bethe代数的极限。我们的主要工具是计算B(C)的幂零退化的Shapovalov型行列式。
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引用次数: 3
Hypergeometric Integrals Modulo p and Hasse–Witt Matrices 模p与Hasse–Witt矩阵的超几何积分
Q3 Mathematics Pub Date : 2020-11-21 DOI: 10.1007/s40598-020-00168-2
Alexey Slinkin, Alexander Varchenko

We consider the KZ differential equations over ({mathbb {C}}) in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field ({mathbb {F}}_p). We study the space of polynomial solutions of these differential equations over ({mathbb {F}}_p), constructed in a previous work by Schechtman and the second author. Using Hasse–Witt matrices, we identify the space of these polynomial solutions over ({mathbb {F}}_p) with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over ({mathbb {F}}_p) and the hypergeometric solutions over ({mathbb {C}}).

当超几何解是一维积分时,我们考虑({mathbb{C}})上的KZ微分方程。我们还考虑有限域上的相同微分方程({mathbb{F}}_p)。我们研究了Schechtman和第二作者先前的工作中构造的这些微分方程在({mathbb{F}}_p)上的多项式解的空间。使用Hasse–Witt矩阵,我们确定了这些多项式解在({mathbb{F}}_p)上的空间,该空间与相关曲线上正则微分的某个子空间对偶。我们还将这些多项式解与({mathbb{F}}_p)上的超几何解联系起来。
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引用次数: 8
A Boothby–Wang Theorem for Besse Contact Manifolds Besse接触流形的Boothby-Wang定理
Q3 Mathematics Pub Date : 2020-11-19 DOI: 10.1007/s40598-020-00165-5
Marc Kegel, Christian Lange

A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal (S^1)-orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition, this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry (Boyer and Galicki in Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008). We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way, we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner–Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism.

如果接触流形上的Reeb流的所有轨道都是周期性的,可能有不同的周期,则称为Besse。我们刻画了Reeb流为Besse的接触流形在满足某些上同调条件的积分辛轨道上的主轨道。除了同调条件外,这一说法还出现在Boyer和Galicki的Sasakian几何语言著作中(Boyer和Gallicki在Sasakia几何中,牛津数学专著,牛津大学出版社,牛津,2008年)。在对上述结果的证明中,我们举例说明了一些不常用的关于轨道的观点。更准确地说,我们通过具有有限稳定群的光滑李群作用,将轨道作为流形的商。通过以这种等价的方式引入所有相关的轨道图概念,我们避免了用轨道图修补结构。作为一个应用,并在Cristofaro Gardiner–Mazzuccelli的工作基础上,我们推导了闭合Besse接触3-流形的一个完整分类,直至严格接触纯性。
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引用次数: 9
A Boothby–Wang Theorem for Besse Contact Manifolds Besse接触流形的一个booth - wang定理
Q3 Mathematics Pub Date : 2020-11-19 DOI: 10.1007/s40598-020-00165-5
M. Kegel, Christian Lange
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引用次数: 0
Torus Action on Quaternionic Projective Plane and Related Spaces 四元数投影平面上的Torus作用及其相关空间
Q3 Mathematics Pub Date : 2020-11-18 DOI: 10.1007/s40598-020-00166-4
Anton Ayzenberg

For an effective action of a compact torus T on a smooth compact manifold X with nonempty finite set of fixed points, the number (frac{1}{2}dim X-dim T) is called the complexity of the action. In this paper, we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that ({mathbb {H}}P^2/T^3cong S^5) and (S^6/T^2cong S^4), for the homogeneous spaces ({mathbb {H}}P^2={{,mathrm{Sp},}}(3)/({{,mathrm{Sp},}}(2)times {{,mathrm{Sp},}}(1))) and (S^6=G_2/{{,mathrm{SU},}}(3)). Here, the maximal tori of the corresponding Lie groups ({{,mathrm{Sp},}}(3)) and (G_2) act on the homogeneous spaces from the left. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of eight-dimensional manifolds with the action of (T^3). This class generalizes ({mathbb {H}}P^2). We prove that their orbit spaces are homeomorphic to (S^5) as well. We link this result to Kuiper–Massey theorem and its generalizations studied by Arnold.

对于紧致环面T在具有非空有限不动点集的光滑紧致流形X上的有效作用,数(frac{1}{2}dim X-dim T)称为作用的复杂性。在本文中,我们研究了复杂一环面作用的某些例子,并描述了它们的轨道空间。我们证明了齐次空间({mathbb{H}}P ^2/T^3cong S^5)和(S^6/T^2 cong S^ 4),对于齐次空间({ mathbb{H}}P ^ 2={{,mathrm{Sp},})/({, mathrm{Sp},)}(2)times}(3))。这里,对应李群({{,mathrm{Sp},}}(3))和(G_2 )的最大托里作用于从左起的齐次空间。接下来我们考虑光滑复曲面的四元数类似物:它们给出了一类具有(T^3)作用的八维流形。这个类推广了({mathbb{H}}P ^2)。我们证明了它们的轨道空间也同胚于(S^5)。我们将这一结果与Arnold研究的Kuiper–Massey定理及其推广联系起来。
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引用次数: 9
Herman Rings of Elliptic Functions 椭圆函数的Herman环
Q3 Mathematics Pub Date : 2020-11-11 DOI: 10.1007/s40598-020-00167-3
Mónica Moreno Rocha

It has been shown by Hawkins and Koss that over any given lattice, the Weierstrass (wp ) function does not exhibit cycles of Herman rings. We show that, regardless of the lattice, any elliptic function of order two cannot have cycles of Herman rings. Through quasiconformal surgery, we obtain the existence of elliptic functions of order at least three with an invariant Herman ring. Finally, if an elliptic function has order (oge 2), then we show there can be at most (o-2) invariant Herman rings.

Hawkins和Koss已经证明,在任何给定的格上,Weierstrass(wp)函数都不表现出Herman环的循环。我们证明,无论晶格如何,任何二阶椭圆函数都不可能有赫尔曼环的环。通过拟共形运算,我们得到了具有不变Herman环的至少三阶椭圆函数的存在性。最后,如果一个椭圆函数具有阶(oge2),则我们证明最多可以存在(o-2)不变的Herman环。
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引用次数: 3
Foreword to the Special Issue Dedicated to Misha Lyubich 米莎·柳比奇特刊前言
Q3 Mathematics Pub Date : 2020-11-11 DOI: 10.1007/s40598-020-00164-6
Anna Miriam Benini, Tanya Firsova, Scott Sutherland, Michael Yampolsky
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引用次数: 0
Fatou’s Associates Fatou的合伙人
Q3 Mathematics Pub Date : 2020-10-26 DOI: 10.1007/s40598-020-00148-6
Vasiliki Evdoridou, Lasse Rempe, David J. Sixsmith

Suppose that f is a transcendental entire function, (V subsetneq {mathbb {C}}) is a simply connected domain, and U is a connected component of (f^{-1}(V)). Using Riemann maps, we associate the map (f :U rightarrow V) to an inner function (g :{mathbb {D}}rightarrow {mathbb {D}}). It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (fU) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain.

假设f是超越整函数,(Vsubsetneq{mathbb{C}})是单连通域,U是(f^{-1}(V))的连通分量。使用黎曼映射,我们将映射(f:Urightarrow V)与内函数(g:{mathbb{D}}rightarrow{math bb{D})相关联。很容易看出,g要么是有限Blaschke乘积,要么通过适当的归一化,可以被视为无限Blaschke积。我们证明,当f在V中的奇异值远离边界时,g的奇异性与U中无穷大的可达性之间存在很强的关系。在U是f的前向不变Fatou分量的情况下,这导致了关于映射g的奇异数的早期结果的非常显著的推广。如果U是f的前向不变Fatou分量,则目前很少有计算对(f,U)和函数g之间关系的例子。我们研究了几个著名的超验整体函数族的这种关系。同样自然地,我们会问哪些有限Blaschke乘积可以以这种方式产生,我们展示了以下内容:对于每一个Julia集与单位圆重合的有限Blaschke-乘积g,都存在一个具有不变Fatou分量的超越整体函数f,使得g在上述意义上与f相关联。此外,存在一个单一的超越整体函数f,其性质是任何有限的Blaschke乘积都可以由与f在游荡域的限制相关的内函数任意逼近。
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引用次数: 3
A Topological Bound on the Cantor–Bendixson Rank of Meromorphic Differentials 亚纯微分的Cantor-Bendixson秩的拓扑界
Q3 Mathematics Pub Date : 2020-10-20 DOI: 10.1007/s40598-020-00163-7
Guillaume Tahar

In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles (corresponding to meromorphic differentials). The Cantor–Bendixson rank of their set of directions is a measure of descriptive set-theoretic complexity. Drawing on a previous work of David Aulicino, we prove a sharp upper bound that depends only on the genus of the underlying topological surface. The proof uses a new geometric lemma stating that in a sequence of three nested invariant subsurfaces the genus of the third one is always bigger than the genus of the first one.

在有限面积的平移曲面(对应于全纯微分)中,鞍连接的方向在单位圆中是稠密的。相反,鞍连接在具有极点的平移曲面中较少(对应于亚纯微分)。它们的方向集的Cantor–Bendixson秩是描述集合论复杂性的度量。根据David Aulicino之前的工作,我们证明了一个仅取决于底层拓扑曲面的亏格的尖锐上界。该证明使用了一个新的几何引理,指出在一个由三个嵌套不变子曲面组成的序列中,第三个子曲面的亏格总是大于第一个子曲面。
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引用次数: 0
期刊
Arnold Mathematical Journal
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