Dimension properties of the boundaries of exponential basins

TL;DR

This paper investigates the fractal geometry of boundaries of exponential basins, showing they have Hausdorff dimension between 1 and 2, with distinct dimensions for escaping and non-escaping points.

Contribution

It establishes precise Hausdorff dimension bounds for boundaries of exponential basins and differentiates between escaping and non-escaping boundary points.

Findings

Boundary Hausdorff dimension > 1 and < 2

Escaping boundary points have Hausdorff dimension 1

Non-escaping boundary points have Hausdorff dimension > 1

Abstract

We prove that the boundary of a component $U$ of the basin of an attracting periodic cycle (of period greater than 1) for an exponential map on the complex plane has Hausdorff dimension greater than 1 and less than 2. Moreover, the set of points in the boundary of $U$ which do not escape to infinity has Hausdorff dimension (in fact: hyperbolic dimension) greater than 1, while the set of points in the boundary of $U$ which escape to infinity has Hausdorff dimension 1.