On The Dynamics Of The Rational Family $\mathbf{f_t(z)=-\frac{t}{4}\frac{(z^{2}-2)^{2}}{z^{2}-1}}$

TL;DR

This paper explores the dynamics and parameter space of a specific family of rational maps, revealing diverse boundary structures of basins at infinity and conditions for Julia sets to be Sierpiński curves.

Contribution

It characterizes the boundary structures of basins at infinity for all escape parameters in a rational family and links these structures to the Julia set topology.

Findings

Boundaries are either Cantor sets, infinitely connected curves, or Jordan curves.

When the boundary is a Jordan curve, the Julia set is a Sierpiński curve.

The paper provides a detailed classification of the parameter space based on boundary types.

Abstract

In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps $\ds f_t(z)=-\frac{t}{4}\frac{(z^{2}-2)^{2}}{z^{2}-1}$ with free critical orbit $\pm\sqrt{2}\xrightarrow{(2)}0\xrightarrow{(4)}t\xrightarrow{(1)}...$. In particular it is shown that for any escape parameter $t$ the boundary of the basin at infinity $\A_t$ is either a Cantor set, a curve with infinitely many complementary components, or else a Jordan curve. In the latter case the Julia set is a Sierpi\'nski curve.