The Dimension Paradox in Parameter Space of Cosine Family

Abstract

It is proved in this paper that the union of escaping parameter rays without endpoints for the cosine family S _κ (z) = e ^κ(e ^z + e^−z), where κ ∈ ℂ is a parameter, has Hausdorff dimension 1, which implies that the ray endpoints alone have Hausdorff dimension 2. This shows that Karpińska’s dimension paradox occurs also in the parameter plane of the cosine family.

Keywords

Dimension paradox / Hausdorff dimension / Parameter rays / Dynamic rays