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

TL;DR

This paper investigates the complex dynamics of the family $f_\lambda(z)=\lambda + z+ an z$, revealing various wandering domains and invariant Baker domains, with detailed behavior depending on the parameter $\lambda$ and its relation to rational or irrational numbers.

Contribution

It classifies wandering domains and invariant Baker domains for the family $f_\lambda(z)$, providing new insights into their topological and dynamical properties based on parameter values.

Findings

Existence of multiple wandering domains with disjoint orbits in the lower half-plane.

Presence of a completely invariant Baker domain containing the upper half-plane.

Different internal behaviors of orbits depending on $\lambda$, including escaping and bounded tendencies.

Abstract

Dynamics of an one-parameter family of functions $f_\lambda(z)=\lambda + z+\tan z, z \in \mathbb{C}$ and $\lambda \in \mathbb{C}$ with an unbounded set of singular values is investigated in this article. For $|2+\lambda^2|<1$, $\lambda=i$, $2+\lambda^2=e^{2\pi i \alpha}$ for some rational number $\alpha$ and for some bounded type irrational number $\alpha$, the dynamics of $f_{\lambda+m\pi}$ is determined for $m \in \mathbb{Z}\setminus\{0\}$. For such values of $\lambda$, the existence of $m$ many wandering domains of $f_{\lambda+m\pi}$ 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 $\{f^n_{\lambda+m\pi}\}$ on such a wandering domain $W$ are highlighted for different values of $\lambda$. More precisely, for $\mid2+\lambda^2\mid<1$, it is manifested that the forward orbit of any point $z\in W$ stays away from the boundaries of $W_n$s. For $\lambda=i$, it is proved that $\liminf_{n\rightarrow \infty}dist(f^n_{i+m\pi}(z),\partial W_n)=0$ for all $z\in W$. Further, $\Im(f^n_{i+m\pi}(z))\rightarrow -\infty$ as $n \rightarrow \infty$. For $2+\lambda^2=e^{2\pi i\alpha}$ for some rational number $\alpha$, $\liminf_{n\rightarrow \infty}dist(f^n_{\lambda+m\pi}(z),\partial W_n)=0$ is established for all $z\in W$. But, $\Im(f^n_{\lambda+m\pi}(z))$ tends to a finite point for all $z\in W$ whenever $n \rightarrow \infty$. For $2+\lambda^2=e^{2\pi i\alpha}$, $\liminf_{n\rightarrow \infty}dist(f^n_{\lambda+m\pi}(z),\partial W_n)>0$ for all $z\in W$ and $dist(f^n_{\lambda+m\pi}(z),f^n_{\lambda+m\pi}(z')=dist(z,z')$ is authenticated for all $z,z'\in W$ and for some bounded type irrational number $\alpha$.