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
References:
[1]

L. Carleson and T. W. Gamelin, "Complex Dynamics,'', Springer Verlag, (1993).   Google Scholar

[2]

M. Comerford, "Properties of Julia Sets for The Arbitrary Composition of Monic Polynomials with Uniformly Bounded Coefficients,'', Ph. D. Thesis, (2001).   Google Scholar

[3]

M. Comerford, A survey of results in random iteration,, Proceedings Symposia in Pure Mathematics, (2004).   Google Scholar

[4]

M. Comerford, Conjugacy and counterexample in random iteration,, Pac. J. of Math., 211 (2003), 69.  doi: 10.2140/pjm.2003.211.69.  Google Scholar

[5]

A. È. Erëmenko and M. J. Lyubich, Examples of entire functions with pathological dynamics,, J. London Math. Soc. (2), 36 (1987), 458.   Google Scholar

[6]

J. E. Fornaess and N. Sibony, Random iterations of rational functions,, Ergodic Theory Dynamical Systems, 11 (1991), 687.  doi: 10.1017/S0143385700006428.  Google Scholar

[7]

Curtis T. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Study 135, (1994).   Google Scholar

[8]

Curtis T. McMullen, Frontiers in complex dynamics,, Bull. Amer. Math. Soc., 31 (1994), 155.   Google Scholar

[9]

R. Ma né, P. Sad and D. Sullivan, On the dynamics of rational maps,, Ann. Sc. de l'Ecole Normale Supérieure, 16 (1983), 193.   Google Scholar

[10]

L. Rempe and S. Van Strien, Absence of line fields and Ma né's theorem for nonrecurrent transcendental functions,, Transactions of the American Mathematical Society, 363 (2011), 203.  doi: 10.1090/S0002-9947-2010-05125-6.  Google Scholar

[11]

Xiaoguang Wang, Rational maps admitting meromorphic invariant line fields,, Bull. Aust. Math. Soc., 80 (2009), 454.  doi: 10.1017/S0004972709000495.  Google Scholar

show all references

References:
[1]

L. Carleson and T. W. Gamelin, "Complex Dynamics,'', Springer Verlag, (1993).   Google Scholar

[2]

M. Comerford, "Properties of Julia Sets for The Arbitrary Composition of Monic Polynomials with Uniformly Bounded Coefficients,'', Ph. D. Thesis, (2001).   Google Scholar

[3]

M. Comerford, A survey of results in random iteration,, Proceedings Symposia in Pure Mathematics, (2004).   Google Scholar

[4]

M. Comerford, Conjugacy and counterexample in random iteration,, Pac. J. of Math., 211 (2003), 69.  doi: 10.2140/pjm.2003.211.69.  Google Scholar

[5]

A. È. Erëmenko and M. J. Lyubich, Examples of entire functions with pathological dynamics,, J. London Math. Soc. (2), 36 (1987), 458.   Google Scholar

[6]

J. E. Fornaess and N. Sibony, Random iterations of rational functions,, Ergodic Theory Dynamical Systems, 11 (1991), 687.  doi: 10.1017/S0143385700006428.  Google Scholar

[7]

Curtis T. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Study 135, (1994).   Google Scholar

[8]

Curtis T. McMullen, Frontiers in complex dynamics,, Bull. Amer. Math. Soc., 31 (1994), 155.   Google Scholar

[9]

R. Ma né, P. Sad and D. Sullivan, On the dynamics of rational maps,, Ann. Sc. de l'Ecole Normale Supérieure, 16 (1983), 193.   Google Scholar

[10]

L. Rempe and S. Van Strien, Absence of line fields and Ma né's theorem for nonrecurrent transcendental functions,, Transactions of the American Mathematical Society, 363 (2011), 203.  doi: 10.1090/S0002-9947-2010-05125-6.  Google Scholar

[11]

Xiaoguang Wang, Rational maps admitting meromorphic invariant line fields,, Bull. Aust. Math. Soc., 80 (2009), 454.  doi: 10.1017/S0004972709000495.  Google Scholar

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