Escaping sets of continuous functions

Hausdorff Dimensions of Escaping Sets of Transcendental Entire Functions

Let f and g be transcendental entire functions, each with a bounded set of singular values, and suppose that g ◦ φ = ψ ◦ f , where φ, ψ : C → C are affine. We show that the escaping sets of f and g have the same Hausdorff dimension. Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equa...

Micro Tangent Sets of Continuous Functions

Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set A we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph...

K -Trivial Closed Sets and Continuous Functions

We investigate the notion of K-triviality for closed sets and continuous functions. Every K-trivial closed set contains a K-trivial real. There exists a K-trivial Π 1 class with no computable elements. For any K-trivial degree d, there is a K-trivial continuous function of degree d.

Γ-sets and Γ-continuous Functions

Connected Escaping Sets of Exponential Maps

We show that, for many parameters a ∈ C, the set I(fa) of points that converge to infinity under iteration of the exponential map fa(z) = e + a is connected. This includes all parameters for which the singular value a escapes to infinity under iteration of fa.