Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts

TL;DR

This paper proves that for certain meromorphic maps with logarithmic tracts, the Julia set contains a hyperbolic Cantor set with Hausdorff dimension exceeding 1, indicating complex fractal structure.

Contribution

It establishes that the hyperbolic dimension of Julia sets for these maps is greater than 1, a novel result in complex dynamics.

Findings

Julia set contains a hyperbolic Cantor set with Hausdorff dimension > 1

Hyperbolic dimension of Julia set exceeds 1 for maps with logarithmic tracts

Results apply to entire and meromorphic maps with finitely many poles in class B

Abstract

We prove that for meromorphic maps with logarithmic tracts (e.g. entire or meromorphic maps with a finite number of poles from class $\mathcal B$), 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.