Linda Keen, Janina Kotus
We study the dynamics of the tangent family z -> lambda tan z for lambda complex and give a complete classification of their stable behavior. We also characterize the the hyperbolic components and give a combinatorial description their deployment in the parameter plane.
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We consider the family of entire transcendental maps given by $F_{λ,m}= λz^m exp(z)$ where m>=2. All functions $F_{λ,m}$ have a superattracting fixed point at z=0, and a critical point at z=-m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behaviour, i.e., λvalues such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then non-locally connected. However, there are parameter values for whi…
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。
Combinatorial structure of the parameter plane of the family $λ\tan z^2$
本文研究了族 $λ\tan z^2$ 参数平面的组合结构，证明了除周期一的四个分支外所有双曲分支均有界且单连通。
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.
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"…
This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_λ=λ\tan^p z^q$. These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are re…
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.
Iteration of the function $f_λ(z)=λ+ z+\tan z, z \in \mathbb{C}$ is investigated in this article. It is proved that for every $λ$, the Fatou set of $f_λ$ has a completely invariant Baker domain $B$; we call it the primary Fatou component. The rest of the results deals with $f_λ$ when it is topologically hyperbolic. For all real $λ$ or $λ$ such that $ λ=πk +i λ_2$ for some integer $k$ and $0 < λ_2<1$, the only other Fatou component is shown to be another completely invariant Baker domain. It is proved that if $|2+λ^2|<1$, then the Fatou set is the union of $B$ and infinitely many invariant attr…
This article discusses some topological properties of the dynamical plane ($z$-plane) of the holomorphic family of meromorphic maps $λ+ \tan z^2$ for $ λ\in \mathbb C$. In the dynamical plane, we prove that there is no Herman ring, and the Julia set is a Cantor set for the maps when the parameter is in the unbounded hyperbolic component contained in the four quadrants in the complex plane. Julia set is connected for the maps when the parameters are in other hyperbolic components of the parameter plane.
Meromorphic Functions with a Polar Asymptotic Value
本文研究一类介于指数族与正切族之间、具有极渐近值的亚纯函数，揭示其动力平面和参数平面中两种已知行为模式的微妙交互。
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。
Dynamics of an one-parameter family of functions $f_λ(z)=λ+ z+\tan z, z \in \mathbb{C}$ and $λ\in \mathbb{C}$ with an unbounded set of singular values is investigated in this article. For $|2+λ^2|<1$, $λ=i$, $2+λ^2=e^{2πi α}$ for some rational number $α$ and for some bounded type irrational number $α$, the dynamics of $f_{λ+mπ}$ is determined for $m \in \mathbb{Z}\setminus\{0\}$. For such values of $λ$, the existence of $m$ many wandering domains of $f_{λ+mπ}$ with disjoint grand orbits in the lower half-plane are asserted along with a completely invariant Baker domain containing the upper hal…
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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