Weiwei Cui, Hongming Nie
We study Siegel disks in the dynamics of functions from the tangent family. In particular, we prove that a forward invariant Siegel disk is unbounded if and only if it contains at least one asymptotic value on the boundary. Our argument is elementary and function-theoretic. Moreover, by using quasiconformal surgery we also construct functions in the above family with bounded Siegel disks.
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For quadratic polynomials of one complex variable, the boundary of the golden-mean Siegel disk must be a quasicircle. We show that the analogous statement is not true for quadratic Hénon maps of two complex variables.
In 2008 Petersen posed a list of questions on the application of trans-quasiconformal Siegel surgery developed by Zakeri and himself. In this paper we extend Petersen-Zakeri's idea so that the surgery can be applied to all the premodels which have no "free critical points". We explain how the idea is used in solving three of the questions posed by Petersen. To present the details of the idea, we focus on the solution of one of them: we prove that for typical rotation numbers $0< θ< 1$, the boundary of the Siegel disk of $f_θ(z) = e^{2 πi θ} \sin (z)$ is a Jordan curve which passes through exac…
As was recently shown by the first author and others, golden-mean Siegel disks of sufficiently dissipative complex quadratic Hénon maps are bounded by topological circles. In this paper we investigate the geometric properties of such curves, and demonstrate that they cannot be $C^1$-smooth.
It was recently shown by Gaidashev and Yampolsky that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal, and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalization in the Hénon family considered by de Carvalho, Lyubich and Martens. As an application of our result, we prove that the boundary of the g…
For the family of quadratic rational functions having a $2$-cycle of bounded type Siegel disks, we prove that each of the boundaries of these Siegel disks contains at most one critical point. In the parameter plane, we prove that the locus for which the boundaries of the $2$-cycle of Siegel disks contain two critical points is a Jordan curve.
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…
An answer to a question of Herman, Baker and Rippon concerning Siegel disks
我们给出了一个肯定的回答赫尔曼，贝克和Rippon的问题，表明，在指数族中，每一个无界西格尔磁盘的边界包含奇异值。
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 prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an indifferent fixed point at $0$, to be parameterized by the rotation number $α$, to depend on $α$ in a Lipschitz-continuous way, and to be non-degenerate. A degenerate family is one for which the set of non-linearizable maps is not dense. We give a characterization of degenerate families, which proves that they are quite exceptional.
We show that the dynamics of sufficiently dissipative semi-Siegel complex Hénon maps with golden-mean rotation number is not $J$-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense $G_δ$ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.
Quadratic Siegel Disks with Rough Boundaries
在二次族（2次多项式的集合）中，Petersen和Zakeri证明了边界是Jordan曲线而不是拟圆的Siegel圆盘的存在。在他们的例子中，临界点包含在曲线中。在第一部分中，我们证明了不含临界点的二次型例子的存在性。在第二部分中，使用一个更抽象的观点（建议阿维拉），我们表明，我们可以控制相当精确的程度的正则性的边界的二次西格尔磁盘，我们创建的扰动。例如，存在边界是C^n但不是C^{n+1}的例子。
We define a new quasiregular mapping T in three dimensions that generalizes the tangent function on the complex plane and shares a number of its geometric properties. We investigate the dynamics of the family λT for λ>0, establishing results analogous to those of Devaney and Keen for the meromorphic family λtan z, λ>0, although the methods used are necessarily original.
We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3. Shishikura's quasiconformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we can transfer McMullen's results on the fine local geometry of Siegel disks to the Herman ring setting.
Dynamics of the family $λ$ tangent $z^2$
本文研究亚纯映射族 λ tan z^2 的动力学平面拓扑性质，证明在含原点的双曲分支中无 Herman 环且 Julia 集为 Cantor 集，在其他双曲分支中 Julia 集连通。
In this paper we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family $\{ T_t(z)=i t\tan z\}_{0< t\leq π}$. Because tangent maps have no critical points but have an essential singularity at infinity and two symmetric asymptotic values, there are new phenomena: as $t$ increases we find single instances of "period quadrupling", "period splitting" and standard "period doubling"; there follows a general pattern of "period merging" where two attracting cycles of period $2^n$ "merge" into one attracting cycle of period $2^{n+1}$, and "cycle doubling"…
Constructing Non-Computable Julia Sets
我们完全刻画了Siegel圆盘在族$$P_θ(z)=e^{2πiθ}z+z^2,$$中对应于{ \bf computable}参数$θ$的共形半径.因此，我们建设性地产生二次多项式与{ \bf不可计算}朱利亚集。
Dynamics of the family lambda tan z
研究了λ复形的正切族z -> λ tan z的动力学，并给出了它们稳定行为的完整分类.我们还刻画了双曲分量的特征，并给出了它们在参数平面上的配置的组合描述。
A universal model for the bifurcations of asymptotic values
本文研究了渐近值分支的通用模型，通过定义Tandelbrot集，证明了Tangent-like映射的直化定理及其参数版本，揭示了多个解析映射族中拓扑Tandelbrot集的存在。
Term structure shapes and their consistent dynamics in the Svensson family
本文研究了Svensson族中远期和收益率曲线的可达形状分类，发现在无套利条件下一致动态会进一步限制可达形状，长期内单一曲线形状必然占主导地位。
Notes on tangent bicharacteristics and ill-posedness of the Cauchy problem
本文构造一族 Hamilton 映射发生谱转变、且双特征可与二重流形相切的二阶双曲算子，并据此证明高阶 Gevrey 类中 Cauchy 问题在原点局部不可解。
We construct the family of algebroid brackets $[\cdot,\cdot]_{c,v}$ on the tangent bundle $T^*M$ to a Poisson manifold $(M,π)$ starting from an algebroid bracket of differential forms. We use these brackets to generate Poisson structures on the tangent bundle $TM$. Next, in the case when $M$ is equipped with a bi-Hamiltonian structure $(M,π_1, π_2)$ we show how to construct another family of Poisson structures. Moreover we present how to find Casimir functions for those structures and we discuss some particular examples.
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