Let g be a polynomial automorphism of C. We study the dimensions (Hausdorff dimension and topological dimension) of the Julia set of g. We show that when g is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly greater than its topological dimension. Moreover, the Julia set cannot be locally connected. We also provide estimates for the dimension of the Julia sets in ...

We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated points and that most zeros are located in neighborhoods of finitely many rays. We apply the result to solutions of linear differential equations.

In the main result of this paper we show that if the Julia set of a meromorphic function f contains a free analytic Jordan arc then it must in fact be a straight line, circle, segment of a straight line or an arc of a circle. If f is transcendental then the Julia set is unbounded and so consists of one or two straight line segments. We construct examples of functions whose Julia sets are of thi...

We analyze the infinitesimal effect of holomorphic perturbations of the dynamics of a structurally stable rational map on a neighborhood of its Julia set. This implies some restrictions on the behavior of critical points.

Abstract. Let φ : PNK → P N K be a morphism of degree d ≥ 2 defined over a field K that is algebraically closed field and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function ĝφ associated to φ is Hölder continuous on P (K) and that the Fatou set F(φ) of φ is equal to the set of points at which ĝΦ is locally constant. Further, ĝφ vanishes precisely o...

For a nitely generated rational semigroup G we establish the existence of a probability measure = G on the Julia set J(G) which has a certain invariance property with respect to the semigroup.

We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. We show that for such a polynomial semigroup, if A and B are two connected components of the Julia set, then one of A and B surrounds the other. A criterion for the Jul...

Makienko's conjecture, a proposed addition to Sul-livan's dictionary, can be stated as follows: The Julia set of a rational function R : C ∞ → C ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko's conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collectio...