On dynamics of $\lambda + \tan z^2 $

Abstract

UDC 517.9

We propose a new family of transcendental meromorphic functions $\lambda  + \tan z^2$ for $ \lambda \in \mathbb C$ and study the dynamics of the family of functions. We explore both the dynamical plane ($z$-plane) and the parameter plane ($\lambda$-plane). We show that, in the dynamical plane, there are no Herman rings, and the Julia set forms a Cantor set when the parameter lies within the unbounded hyperbolic components. In addition, it is proved that these unbounded hyperbolic components are the only available components  distributed over the four quadrants of the parameter space in the complex plane. Conversely, it is shown that the Julia set is connected for the maps whose  parameter lies within the remaining hyperbolic components of the parameter space. We also perform the comprehensive analysis of the combinatorial structure of both the parameter space and the dynamical plane for this family of transcendental meromorphic maps.