Julia sets converging to the unit disk

Abstract

We consider the family of rational maps F λ ( z ) = z n + λ / z d F_\lambda (z) = z^n + \lambda /z^d , where n , d ≥ 2 n,d \geq 2 and λ \lambda is small. If λ \lambda is equal to 0, the limiting map is F 0 ( z ) = z n F_0(z)=z^n and the Julia set is the unit circle. We investigate the behavior of the Julia sets of F λ F_\lambda when λ \lambda tends to 0, obtaining two very different cases depending on n n and d d . The first case occurs when n = d = 2 n=d=2 ; here the Julia sets of F λ F_\lambda converge as sets to the closed unit disk. In the second case, when one of n n or d d is larger than 2 2 , there is always an annulus of some fixed size in the complement of the Julia set, no matter how small | λ | |\lambda | is.