Geometric limits of Julia  sets and connectedness locus of the family of polynomials Pc(z) = zn + czk. Issue 4 (2nd October 2019)

• Record Type: Journal Article
• Authors: Alves, Alexandre Miranda
• Is Part Of: Dynamical systems. Volume 34:Issue 4(2019)
• Page Start: 668
• Page End: 684
• Publication Date: 2019-10-02
• Subjects: Complex dynamics -- geometric limits -- connectedness locus -- Julia set; 37F45 -- 37F40; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2019.1593321
• Languages: English
• ISSNs: 1468-9367

Abstract

ABSTRACT: Let n and k be positive integers. For n > k ≥ 2, we consider the connectedness locus M n, k of the family of polynomials P c ( z ) = z n + c z k, where c is a complex parameter. We show that the geometric limit of the connectedness locus sets M n, k, when n tends to infinity, exists and is the closed unit disk. In addition, we give an upper bound for the geometric size of M n, k .When parameter c belongs to the open unit disk, we show that the geometric limit of the Julia sets J c, when n tends to infinity, exists and is the unit circle. Finally, we have established some properties on the hyperbolic components of this family.