论文标题
家庭中不同类型的流浪域$λ+z+\ tan z $
Different types of wandering domains in the family $ λ+z+\tan z$
论文作者
论文摘要
单参数家族的动力学$f_λ（z）=λ+ z+ z+ \ tan z，z \ in \ mathbb {c} $和$λ\ in \ mathbb {c} $ in \ mathbb {c} $，在本文中研究了一组无绑的单数值集。对于$ | 2+λ^2 | <1 $，$λ= i $，$ 2+λ^2 = e^{2πiα} $对于某些有理数$α$，对于某些有界类型的非理性数字$α$，$ f_ {λ+mπ} $的动态是$ f_ {λ+mπ} $的动态，用于$ m \ in \ mathbb in \ mathbb} $ {z} $ {z} $ {Z} $ {0 c} $ {0 c} $ {0。对于$λ$的这样的值，$ m $的存在$ f_ {λ+mπ} $的许多流浪域与下半平面中的脱节大轨道的存在与一个完全不变的面包师域，其中包含上半平面。此外，发现每个流浪的域都简单地连接，无限制和逃脱。 $ \ {f^n_ {λ+mπ} \} $的不同类型的内部行为在这样的流浪域上$ w $突出显示了$λ$的不同值。更确切地说，对于$ \ mid2+λ^2 \ mid <1 $，可以表明，w $中任何点$ z \的前向轨道都远离$ w_n $ s的边界。对于$λ= i $，证明$ \ liminf_ {n \ rightArrow \ infty} dist（f^n_ {i+mπ}（z）（z），\ partial w_n）= 0 $ for W $中的$ z \。此外，$ \ im（f^n_ {i+mπ}（z））\ rightarrow -\ infty $ as $ n \ rightarrow \ infty $。对于$ 2+λ^2 = e^{2πiα} $对于某些有理数$α$，$ \ liminf_ {n \ rightarrow \ infty} dist（f^n_ {λ+mπ}（z）（z），\ partial w_n）= 0 $在w $中建立了$ z。但是，$ \ im（f^n_ {λ+mπ}（z））$趋向于w $中的所有$ z \ in时，只要$ n \ rightarrow \ rightarrow \ infty $。对于$ 2+λ^2 = e^{2πiα} $，$ \ liminf_ {n \ rightarrow \ infty \ infty} dist（f^n_ {λ+mπ}（z），\ partial w_n）> 0 $ dist（f^n_ {λ+mπ}（z），f^n_ {λ+mπ}（z'）= dist（z，z'）$均为所有$ z，z'\在w $中以及某些有界类型的不合理数量$α$ persin验证。
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 half-plane. Further, each of such wandering domains is found to be simply connected, unbounded, and escaping. Different types of the internal behavior of $\{f^n_{λ+mπ}\}$ on such a wandering domain $W$ are highlighted for different values of $λ$. More precisely, for $\mid2+λ^2\mid<1$, it is manifested that the forward orbit of any point $z\in W$ stays away from the boundaries of $W_n$s. For $λ=i$, it is proved that $\liminf_{n\rightarrow \infty}dist(f^n_{i+mπ}(z),\partial W_n)=0$ for all $z\in W$. Further, $\Im(f^n_{i+mπ}(z))\rightarrow -\infty$ as $n \rightarrow \infty$. For $2+λ^2=e^{2πiα}$ for some rational number $α$, $\liminf_{n\rightarrow \infty}dist(f^n_{λ+mπ}(z),\partial W_n)=0$ is established for all $z\in W$. But, $\Im(f^n_{λ+mπ}(z))$ tends to a finite point for all $z\in W$ whenever $n \rightarrow \infty$. For $2+λ^2=e^{2πiα}$, $\liminf_{n\rightarrow \infty}dist(f^n_{λ+mπ}(z),\partial W_n)>0$ for all $z\in W$ and $dist(f^n_{λ+mπ}(z),f^n_{λ+mπ}(z')=dist(z,z')$ is authenticated for all $z,z'\in W$ and for some bounded type irrational number $α$.
