  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.
