A Straightening Theorem for non-Autonomous Iteration

TL;DR

This paper extends the classical straightening theorem to non-autonomous iteration, establishing conditions under which a sequence of varying functions is hybrid equivalent to a polynomial, using new techniques for distortion control.

Contribution

It generalizes the straightening theorem to non-autonomous settings with variable functions, employing Caratheodory topology for better distortion estimates.

Findings

Established a non-autonomous straightening theorem.

Developed new distortion control techniques.

Applied Caratheodory topology for bounds on polynomial-like sequences.

Abstract

The classical straightening theorem as proved by Douady and Hubbard shows that a polynomial-like sequence is hybrid equivalent to a polynomial. We generalize this result to non-autonomous iteration where one considers composition sequences arising from a varying sequence of functions. In order to do this, new techniques are required to control the distortion and quasiconformal dilatation of the hybrid equivalence. In particular, the Caratheodory topology for pointed domains allows us to specify the appropriate bounds on the sequence of sets on which the polynomial-like mapping sequence is defined and give us good estimates on the degree of distortion and quasiconformality.