Different types of wandering domains in the family $ λ+z+\tan z$

Abstract

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 $α$.