Bernoulli measure of complex admissible kneading sequences

TL;DR

This paper proves that the set of admissible kneading sequences for complex quadratic polynomials has positive Bernoulli measure, indicating a significant size within the space of all possible sequences.

Contribution

The authors establish that the set of admissible kneading sequences in complex dynamics has positive Bernoulli measure, extending prior real case results to the complex setting.

Findings

Admissible kneading sequences form a set of positive Bernoulli measure.

The result generalizes the understanding of kneading sequences from real to complex quadratic polynomials.

Provides a measure-theoretic perspective on the complexity of dynamical systems.

Abstract

Iterated quadratic polynomials give rise to a rich collection of different dynamical systems that are parametrized by a simple complex parameter $c$. The different dynamical features are encoded by the \emph{kneading sequence} which is an infinite sequence over $\{0,\1\}$. Not every such sequence actually occurs in complex dynamics. The set of admissible kneading sequences was described by Milnor and Thurston for real quadratic polynomials, and by the authors in the complex case. We prove that the set of admissible kneading sequences has positive Bernoulli measure within the set of sequences over $\{0,\1\}$.