Rational map has a Cantor Circle Julia set and a Sierpinski Carpet Julia set of degree m

Abstract

This research aims to study complex dynamics, and one of the best examples of this concept is the Julia set, for this we will present new concepts specific to the Julia set of the rational map J(Gμ). We introduce the family of complex rational maps as: Gμ(z) = 2μm+1 z–m + z2m – μ3m+1 / zm (z2m – μm–1) , with m ≥ 2 and μ ϵ ℂ\{0}(ℂ complex numbers) such that μm+1 ≠ 1 and μ2m+2 ≠ 1. We prove J(Gμ) is a degenerate Sierpinski carpet or a Sierpinski carpet or a Cantor circle by condition one of the free critical points has an image for Gμ is diverge on ∞ or 0.