Let f be a polynomial endomorphism of degree d ≥ 2 of C k (k ≥ 2) which extends to a holomorphic endomorphism of P k. Assume that the maximal order Julia set of f is laminated by real hypersurfaces in some open set. We show that f is homogenous and is a polynomial lift of a Lattès endomorphism of P k−1 .

This paper is based on the author’s previous work [S4]. We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a com...

We study a family of rational maps of the sphere with the property that each map has two fixed points with multiplier −1; moreover each map has no period 2 orbits. The family we analyze is Ra(z) = z3−z −z2+az+1 , where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set...

Let f : C → C be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set F(f) and the postsingular set P (f) is compact and the intersection of the Julia set J (f) and P (f) is finite. Assume that no asymptotic value of f belongs to J (f) and that the local degree of f at all points in J (f) is bounded by some finite constant. We prove that there is a hyperbol...

We construct a transcendental entire function whose Julia set has locally finite 1-dimensional measure, hence Hausdorff dimension 1. Date: Jan 5, 2011. 1991 Mathematics Subject Classification. Primary: Secondary:

It is known that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high.

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is bounded and the map f restricted to its closure is expanding, the property refered to as subexpanding. We first show the existence, uniqueness, conservativity ...

DOUGLAS N. HENRY,1,2 JULIA V. BUSIK,1 FRANK C. BROSIUS III,3 AND CHARLES W. HEILIG4 1Department of Physiology, 2Department of Pediatrics and Human Development, Division of Pediatric Endocrinology, College of Human Medicine, Michigan State University, East Lansing 48824-1101; 3Department of Medicine, Division of Nephrology, University of Michigan Medical School and Ann Arbor Veterans Affairs Hos...

This paper contrastively researches the structural characteristic and the fission-evolution law of four different kinds of generalized Julia set generalized J set in short with different parameter c, which includes the generalized J set without any perturbation, the generalized J set perturbed by additive noises, the generalized J set perturbed by multiplicative noise, and the generalized J set...

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.