No invariant line fields on Cantor Julia sets

TL;DR

This paper proves that rational maps with Cantor Julia sets do not admit invariant line fields, implying such maps are hyperbolic if they are structurally stable, thereby advancing understanding of complex dynamics and stability.

Contribution

The paper establishes the non-existence of invariant line fields on Cantor Julia sets for rational maps, linking structural stability to hyperbolicity in this context.

Findings

No invariant line fields on Cantor Julia sets of rational maps.

Structurally stable rational maps with Cantor Julia sets are hyperbolic.

Advances understanding of stability and dynamics in complex rational maps.

Abstract

In this paper, we prove that a rational map with a Cantor Julia set carries no invariant line fields on its Julia set. It follows that a structurally stable rational map with a Cantor Julia set is hyperbolic.