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