On Dynamics of \lambda + tan z^2

TL;DR

This paper explores the topological dynamics of the family of meromorphic maps  + an z^2, revealing the structure of Julia sets and the absence of Herman rings depending on parameter regions.

Contribution

It provides new insights into the topological properties of  + an z^2, including the nature of Julia sets and the existence of Herman rings across different hyperbolic components.

Findings

Julia set is a Cantor set for parameters in certain hyperbolic components.

Julia set is connected for parameters in other hyperbolic components.

No Herman rings exist for these maps in the studied parameter regions.

Abstract

This article discusses some topological properties of the dynamical plane ($z$-plane) of the holomorphic family of meromorphic maps $\lambda + \tan z^2$ for $ \lambda \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.