Preservation of External Rays in non-Autonomous Iteration

TL;DR

This paper studies how external rays and Julia sets behave under non-autonomous polynomial iteration within hyperbolic components, showing their properties are preserved and move holomorphically as parameters vary.

Contribution

It extends the understanding of Julia set dynamics to non-autonomous iteration, demonstrating preservation and holomorphic motion of external rays and Julia sets within hyperbolic components.

Findings

External rays and Julia sets move holomorphically within hyperbolic components.

External rays with fixed angles separate Julia sets consistently across parameters.

Holomorphic motion of Julia sets coincides with grand orbit methods.

Abstract

We consider the dynamics arising from the iteration of an arbitrary sequence of polynomials with uniformly bounded degrees and coefficients and show that, as parameters vary within a single hyperbolic component in parameter space, certain properties of the corresponding Julia sets are preserved. In particular, we show that if the sequence is hyperbolic and all the Julia sets are connected, then the whole basin at infinity moves holomorphically. This extends also to the landing points of external rays and the resultant holomorphic motion of the Julia sets coincides with that obtained earlier using grand orbits. In addition, if a finite set of external rays separate the Julia set for a particular parameter value, then the rays with the same external angles separate the Julia set for every parameter in the same hyperbolic component.