Dierk Schleicher

Escaping Points of Exponential Maps

The points which converge to ∞ under iteration of the maps z↦λexp(z) for λ ∈ C/{0} are investigated. A complete classification of such ‘escaping points’ is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter λ. It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpinska for specific choices of λ.

Hausdorff Dimension, Its Properties, and Its Surprises

We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension has some surprising properties: we construct a set

ON MULTICORNS AND UNICORNS I: ANTIHOLOMORPHIC DYNAMICS, HYPERBOLIC COMPONENTS AND REAL CUBIC POLYNOMIALS

E\subset\C

Exponential Thurston maps and limits of quadratic differentials

of positive planar measure and with dimension 2 such that each point in

Dynamics of Entire Functions

E

Bifurcations in the space of exponential maps

can be joined to

Classification of escaping exponential maps

\infty

Embedding non-Euclidean color spaces into Euclidean color spaces with minimal isometric disagreement

by one or several curves in

RATIONAL PARAMETER RAYS OF THE MULTIBROT SETS

\C

On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials

such that all curves are disjoint from each other and from