We prove several essential fractal properties, such as positivity, finiteness or local infinity, of Hausdorff and packing measures of radial Julia sets of large subclasses of entire and meromorphic functions considered in [MU]. Most of the results proven are shown to be optimal.

In this paper, the dynamics of the functions [Formula: see text] with the real parameter is studied. We say that a real parameter [Formula: see text] belongs to the set [Formula: see text] for a positive integer n if [Formula: see text] has an attracting cycle of n-order. We prove that the Fatou set [Formula: see text] is a completely invariant attracting basin for every parameter [Formula: see...

Let F be a polynomial endomorphism of C which extends holomorphically to CP. We prove that the dimension of μF , the pluriharmonic measure on the boundary of the filled Julia set of F , is bounded above by 2n− 1.

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. Furthermore, we investigate the fiberwise dynamics of skew products related to polynomial semigroups with bounded planar postcritical set. Using uniform fiberwise quasi...

For f an entire transcendental map with univalent Baker domain U of hyperbolic type I, we study pinching deformations in U , the support of this deformation being a lamination in the grand orbit of U . We show that pinching along a lamination that contains the geodesic λ∞ (See Section 3.1) does not converges. However, pinching at a lamination that does not contains such λ∞, converges and conver...

In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impr...

We develop the elementary theory of iterated rational functions over the Riemann sphere C∞ in a constructive setting. We use Bishop-style constructive proof methods throughout. Starting from the development of constructive complex analysis presented in [Bishop and Bridges 1985], we give constructive proofs of Montel’s Theorem along with necessary generalisations, and use them to prove elementar...

0 lies on a periodic orbit. We then perturb F0 by adding a pole at the origin. Our goal is to investigate the structure of the Julia set of Fλ, which we denote by J(Fλ), when λ is nonzero. For these maps, the point at ∞ is always a superattracting fixed point, so we have an immediate basin of attraction of ∞ that we denote by Bλ. As a consequence, we may also define the filled Julia set for the...

Let f : C → Ĉ be a transcendental meromorphic function. Suppose that the finite part P(f)∩C of the postsingular set of f is bounded, that f has no recurrent critical points or wandering domains, and that the degree of pre-poles of f is uniformly bounded. Then we show that f supports no invariant line fields on its Julia set. We prove this by generalizing two results about rational functions to ...