Dynamics of the family $\lambda$ tangent $z^2$

TL;DR

This paper explores the topological dynamics of the family of meromorphic maps λ tan z^2, showing the absence of Herman rings, the Cantor set structure of Julia sets in certain parameters, and connectedness in others.

Contribution

It provides new insights into the topological properties of λ tan z^2 maps, including the structure of Julia sets and the absence of Herman rings in specific hyperbolic components.

Findings

Julia set is a Cantor set for parameters in the hyperbolic component containing the origin.

Julia set is connected for parameters in other hyperbolic components.

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

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, I prove that there is no Herman ring and the Julia set is a Cantor set for the maps when the parameter is in the hyperbolic component containing the origin. Julia set is connected for the maps when the parameters are in other hyperbolic components in the parameter plane.