# 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
##### 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
 [1] Mark Comerford, Todd Woodard. Orbit portraits in non-autonomous iteration. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2253-2277. doi: 10.3934/dcdss.2019144 [2] Barbara Bianconi, Francesca Papalini. Non-autonomous boundary value problems on the real line. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 759-776. doi: 10.3934/dcds.2006.15.759 [3] Cung The Anh, Tang Quoc Bao. Dynamics of non-autonomous nonclassical diffusion equations on $R^n$. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1231-1252. doi: 10.3934/cpaa.2012.11.1231 [4] Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743 [5] Wen Tan, Chunyou Sun. Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6035-6067. doi: 10.3934/dcds.2017260 [6] Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801 [7] Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181 [8] Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703 [9] Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108 [10] Xinguang Yang, Baowei Feng, Thales Maier de Souza, Taige Wang. Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 363-386. doi: 10.3934/dcdsb.2018084 [11] Yun Lan, Ji Shu. Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934/cpaa.2019109 [12] Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3717-3747. doi: 10.3934/dcds.2019151 [13] Iacopo P. Longo, Sylvia Novo, Rafael Obaya. Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5491-5520. doi: 10.3934/dcds.2019224 [14] Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3453-3474. doi: 10.3934/dcdsb.2018251 [15] Thorsten Hüls, Yongkui Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 79-99. doi: 10.3934/dcdsb.2012.17.79 [16] Thorsten Hüls. A model function for non-autonomous bifurcations of maps. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 351-363. doi: 10.3934/dcdsb.2007.7.351 [17] Maciej J. Capiński, Piotr Zgliczyński. Covering relations and non-autonomous perturbations of ODEs. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 281-293. doi: 10.3934/dcds.2006.14.281 [18] Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 [19] Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067 [20] Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701

2018 Impact Factor: 1.143