Krzysztof Barański

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.

Hausdorff dimension of hairs and ends for entire maps of finite order

We study transcendental entire maps f of finite order, such that all the singularities of f-1 are contained in a compact subset of the immediate basin B of an attracting fixed point of f. Then the Julia set of f consists of disjoint curves tending to infinity (hairs), attached to the unique point accessible from B (endpoint of the hair). We prove that the Hausdorff dimension of the set of endpoints of the hairs is equal to 2, while the union of the hairs without endpoints has Hausdorff dimension 1, which generalizes the result for exponential maps. Moreover, we show that for every transcendental entire map of finite order from class (i.e. with bounded set of singularities) the Hausdorff dimension of the Julia set is equal to 2.

On realizability of branched coverings of the sphere

Abstract We introduce a simple geometric method of determining which abstract branch data can be realized by branched coverings of the two-dimensional sphere S 2 (the Hurwitz problem). Using this method we show the realizability of some classes of data.

Univalent Baker domains

We classify Baker domains U for entire maps with f|U univalent into three different types, giving several criteria which characterize them. Some new examples of such domains are presented, including a domain with disconnected boundary in and a domain which spirals towards infinity.

Brushing the hairs of transcendental entire functions

Abstract Let f be a transcendental entire function of finite order in the Eremenko–Lyubich class B (or a finite composition of such maps), and suppose that f is hyperbolic and has a unique Fatou component. We show that the Julia set of f is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if f ∈ B has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic with connected Fatou set, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function f ∈ B , a natural compactification of the dynamical plane by adding a “circle of addresses” at infinity.

On the dimension of the graph of the classical Weierstrass function

Abstract This paper examines dimension of the graph of the famous Weierstrass non-differentiable function W λ , b ( x ) = ∑ n = 0 ∞ λ n cos ⁡ ( 2 π b n x ) for an integer b ≥ 2 and 1 / b λ 1 . We prove that for every b there exists (explicitly given) λ b ∈ ( 1 / b , 1 ) such that the Hausdorff dimension of the graph of W λ , b is equal to D = 2 + log ⁡ λ log ⁡ b for every λ ∈ ( λ b , 1 ) . We also show that the dimension is equal to D for almost every λ on some larger interval. This partially solves a well-known thirty-year-old conjecture. Furthermore, we prove that the Hausdorff dimension of the graph of the function f ( x ) = ∑ n = 0 ∞ λ n ϕ ( b n x ) for an integer b ≥ 2 and 1 / b λ 1 is equal to D for a typical Z -periodic C 3 function ϕ.

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.

ON THE CONNECTIVITY OF THE JULIA SETS OF MEROMORPHIC FUNCTIONS

We prove that every transcendental meromorphic map

On the dimension of graphs of Weierstrass-type functions with rapidly growing frequencies

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