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

Mark Comerford

doi:10.3934/dcds.2013.33.629

Article Contents

2013, Volume 33, Issue 2: 629-642. Doi: 10.3934/dcds.2013.33.629

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

Mark Comerford^1,

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

Received: April 30, 2011

Revised: June 30, 2012

Published: January 2013

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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 with at most countably many points removed.

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Mathematics Subject Classification: Primary: 30D05; Secondary: 37F10.

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References

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