We study the family of complex maps given by F λ (z) = z n + λ/z n + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components.

Abstract. In this paper we consider analytic planar differential systems having a first integral of the form H(x, y) = A(x) + B(x)y + C(x)y 2 and an integrating factor κ(x) not depending on y. Our aim is to provide tools to study the period function of the centers of this type of differential system and to this end we prove three results. Theorem A gives a characterization of isochronicity, a criterion to bound the number of critical periods and a necessary condition for the period function to be monotone. Theorem B is intended for being applied in combination with Theorem A in an algebraic setting that we shall specify. Finally, Theorem C is devoted to study the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers. Four different applications are given to illustrate these results.

We consider the quadratic family of complex maps given by q c (z) = z 2 + c where c is the center of a hyperbolic component in the Mandelbrot set. Then, we introduce a singular perturbation on the corresponding bounded superattracting cycle by adding one pole to each point in the cycle. When c = −1 the Julia set of q −1 is the well known basilica and the perturbed map is given by f λ (z) = z 2 − 1 + λ/ z d 0 (z + 1) d 1 where d 0 , d 1 ≥ 1 are integers, and λ is a complex parameter such that |λ| is very small. We focus on the topological characteristics of the Julia and Fatou sets of f λ that arise when the parameter λ becomes nonzero. We give sufficient conditions on the order of the poles so that for small λ the Julia sets consist of the union of homeomorphic copies of the unperturbed Julia set, countably many Cantor sets of concentric closed curves, and Cantor sets of point components that accumulate on them.

This paper studies the differential equation ż = f (z), where f is an analytic function in C except, possibly, at isolated singularities. We give a unify treatment of well known results and provide new insight into the local normal forms and global properties of the solutions for this family of differential equations.