We show that if a meromorphic function has two completely invariant Fatou components and only finitely many critical and asymptotic values, then its Julia set is a Jordan curve. However, even if both domains are attracting basins, the Julia set need not be a quasicircle. We also show that all critical and asymptotic values are contained in the two completely invariant components. This need not ...

Part I of this paper has been devoted to properties of the different Julia set configurations, generated by the complex map TZ : z′ = z − c, c being a real parameter, −1/4 < c < 2. These properties were revisited from a detailed knowledge of the fractal organization (called “boxwithin-a-box”), generated by the map x′ = x − c with x a real variable. Here, the second part deals with an embedding ...

We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen’s formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition.

In this paper we describe several new types of invariant sets that appear in the Julia sets of the complex exponential functions Eλ(z) = λe z where λ ∈ C. These invariant sets consist of points that share the same itinerary under iteration of Eλ. Since these exponential functions are 2πi periodic, there are several “natural” ways (described below) to decompose the plane into countably many stri...

Let f be an entire function whose set of singular values is bounded and suppose that f has a Siegel disk U such that f |∂U is a homeomorphism. We extend a theorem of Herman by showing that, if the rotation number of U is diophantine, then ∂U contains a critical point of f . More generally, we show that, if U is a (not necessarily diophantine) Siegel disk as above, then U is bounded. Suppose fur...

During early 1980s, the so-called ‘escape time’ method, developed to display the Julia sets for complex dynamical systems, was exported to quaternions in order to draw analogous pictures in this wider numerical field. Despite of the fine results in the complex plane, where all topological configurations of Julia sets have been successfully displayed, the ‘escape time’ method fails to render pro...

We consider the basilica Julia set of the polynomial P (z) = z − 1 and construct all possible resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the effective resistance metric coincides with the usual topology. Then we concentrate on two particular cases. One is a self-similar harmonic structure, for which the energy renormalization factor is 2, the expon...

In this paper, we discuss several aspects of Julia sets, as well as those of the Mandelbrot set. We are interested in topological properties such as connectivity and local connectivity, geometric properties such as Hausdorff dimension and Lebesgue measure, and complex analytic properties such as holomorphic removability. As one can easily see from the pictures of numerical experiments, there is...

We partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider intere...

It is shown that the boundary of the Mandelbrot set M has Hausdorff dimension two and that for a generic c ∈ ∂M , the Julia set of z 7→ z + c also has Hausdorff dimension two. The proof is based on the study of the bifurcation of parabolic periodic points.