Bogusława Karpińska

Hausdorff dimension of the hairs without endpoints for λ exp z

Abstract We consider the complex exponential maps Eλ (z) = λ ez where z ∈ ℂ and λ e (0, 1/e). It is known that the Julia set of Eλ is a Cantor bouquet of curves (“hairs”) extending from the set of their endpoints to ∞. We prove that the Hausdorff dimension of the set of these curves without the endpoints is equal to 1; in particular, it is smaller than the Hausdorff dimension of the set of endpoints alone (which is known to be equal to 2, see [5]).

Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts

We prove that for meromorphic maps with logarithmic tracts (in particular, for transcendental maps in the class , which are entire or meromorphic with a finite number of poles), the Julia set contains a compact invariant hyperbolic Cantor set of Hausdorff dimension greater than 1. Hence, the hyperbolic dimension of the Julia set is greater than 1.

The growth rate of an entire function and the Hausdorff dimension of its Julia set

Let f be a transcendental entire function in the Eremenko�Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.

On the Hausdorff dimension of the Julia set of a regularly growing entire function

We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.

Coding trees and boundaries of attracting basins for some entire maps

Let f be an entire transcendental map, such that all the singularities of f−1 are contained in a compact subset of the immediate basin B(z0) of an attracting fixed point z0. We study the structure of the Julia set of f, which is equal to the boundary of B(z0), and the behaviour of the Riemann mapping onto B(z0) using the technique of geometric coding trees of preimages of points from B(z0). We show that for a given symbolic itinerary, if codes of the tracts of f are bounded and codes of the fundamental domains grow no faster than the iterates of an exponential function, then there exists a point in the Julia set with this itinerary. Moreover, we determine cluster sets for and show that has an unrestricted limit equal to ∞ at points of a dense uncountable set in the unit circle.

How Points Escape to Infinity Under Exponential Maps

We investigate the finer fractal structure of the set of points escaping to infinity under iteration of an arbitrary exponential map. Providing exact formulas, we show how sensitively the Hausdorff dimension depends on the rate of growth of canonical Devaney-Krych codes.

ON THE CONNECTIVITY OF THE JULIA SETS OF MEROMORPHIC FUNCTIONS

We prove that every transcendental meromorphic map

Accesses to infinity from Fatou components

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