Quasiconformal non-equivalence of connected Julia sets and Kleinian limit sets

Prove that for any connected Julia set J of a rational map and any connected limit set Λ of a Kleinian group, provided neither J nor Λ is homeomorphic to a circle or a sphere, J is not quasiconformally homeomorphic to Λ.

Background

Classifying fractal sets up to quasiconformal homeomorphisms is a central question in quasiconformal geometry. For fractals arising in conformal dynamics, substantial evidence suggests Julia sets and Kleinian limit sets should be distinguishable up to quasiconformal equivalence.

The paper establishes strong partial results toward this goal: it proves that no Julia set is quasiconformally homeomorphic to the Apollonian gasket (Theorem A) and that no Julia set of a quadratic rational map is quasiconformally homeomorphic to a gasket limit set of a geometrically finite Kleinian group (Theorem B). Additionally, it develops structural constraints on fat gasket Julia sets (Theorem C and Theorem D) to support these distinctions. The conjecture stated below encapsulates the broader non-equivalence claim beyond these special cases.