Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials

TL;DR

This paper demonstrates that kneading sequences fully encode Hubbard trees and their dynamics for quadratic polynomials, providing a method to reconstruct trees and classify which sequences occur in complex dynamics.

Contribution

It proves the complete encoding of Hubbard trees by kneading sequences and introduces a complex admissibility condition for real and complex dynamics.

Findings

Kneading sequences determine Hubbard trees and their local dynamics.

Not all kneading sequences are realizable in complex dynamics due to admissibility constraints.

The paper characterizes when a kneading sequence corresponds to a realizable Hubbard tree.

Abstract

Hubbard trees are invariant trees connecting the points of the critical orbits of postcritically finite polynomials. Douady and Hubbard \cite{Orsay} introduced these trees and showed that they encode the essential information of Julia sets in a combinatorial way. The itinerary of the critical orbit within the Hubbard tree is encoded by a (pre)periodic sequence on $\{\0,\1\}$ called \emph{kneading sequence}. We prove that the kneading sequence completely encodes the Hubbard tree and its dynamics, and we show how to reconstruct the tree and in particular its branch points (together with their periods, their relative posititions, their number of arms and their local dynamics) in terms of the kneading sequence alone. Every kneading sequence gives rise to an abstract Hubbard tree, but not every kneading sequence occurs in real dynamics or in complex dynamics. Milnor and Thurston \cite{MT} classified which kneading sequences occur in real dynamics; we do the same for complex dynamics in terms of a complex \emph{admissibility condition}. This complex admissibility condition fails if and only if the abstract Hubbard tree has a so-called \emph{evil} periodic branch point that is incompatible with local homeomorphic dynamics on the plane.