# American Institute of Mathematical Sciences

February  2013, 33(2): 629-642. doi: 10.3934/dcds.2013.33.629

## Non-autonomous Julia sets with measurable invariant sequences of line fields

 1 Department of Mathematics，University of Rhode Island, 5 Lippitt Road, Room 102F, Kingston, RI 02881, United States

Received  May 2011 Revised  July 2012 Published  September 2012

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 with at most countably many points removed.
Citation: Mark Comerford. Non-autonomous Julia sets with measurable invariant sequences of line fields. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 629-642. doi: 10.3934/dcds.2013.33.629
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