Non-Autonomous Julia Sets with Invariant Sequences of Measurable Line Fields

TL;DR

This paper demonstrates that in non-autonomous complex dynamics, sequences of quadratic polynomials can have Julia sets with positive area supporting invariant measurable line fields, contradicting the classical no invariant line fields conjecture.

Contribution

It extends the no invariant line fields conjecture to non-autonomous iteration, showing it does not hold in this broader setting by explicit construction.

Findings

Constructed quadratic polynomial sequences with Julia sets of positive area

Established existence of invariant measurable line fields supported on these Julia sets

Contradicted the classical no invariant line fields conjecture in non-autonomous dynamics

Abstract

The no invariant line fields conjecture is one of the main outstanding problems in traditional complex dynamics. In this paper we consider non-autonomous iteration where one works with compositions of sequences of polynomials with suitable bounds on the degrees and coefficients. We show that the natural generalization of the no invariant line fields conjecture to this setting is not true. In particular, we construct a sequence of quadratic polynomials whose iterated Julia sets all have positive area and which has an invariant sequence of measurable line fields whose supports are these iterated Julia sets.