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July  2011, 29(3): 1205-1244. doi: 10.3934/dcds.2011.29.1205

Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets

1. 

Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Received  October 2009 Revised  August 2010 Published  November 2010

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected components of the Julia set, then one of $A$ and $B$ surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the cardinality of the set of all connected components of the Julia set of such a semigroup is given. By using this, we give a criterion for the Julia set to be connected. Moreover, we show that for any $n\in N \cup \{ \aleph _{0}\} ,$ there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all connected components of the Julia set is equal to $n.$ Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomials are found and systematically investigated.
Citation: Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1205-1244. doi: 10.3934/dcds.2011.29.1205
References:
[1]

L. V. Ahlfors, "Conformal Invariants: Topics in Geometric Function Theory," McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Dusseldorf-Johannesburg, 1973.

[2]

A. F. Beardon, Symmetries of Julia sets, Bull. London Math. Soc., 22 (1990), 576-582. doi: 10.1112/blms/22.6.576.

[3]

A. F. Beardon, "Iteration of Rational Functions," Graduate Text of Mathematics 132, Springer-Verlag, 1991.

[4]

R. Brück, Geometric properties of Julia sets of the composition of polynomials of the form $z^{2}+c_n$, Pacific J. Math., 198 (2001), 347-372. doi: 10.2140/pjm.2001.198.347.

[5]

R. Brück, M. Büger and S. Reitz, Random iterations of polynomials of the form $z^{2}+c_n$: Connectedness of Julia sets, Ergodic Theory Dynam. Systems, 19 (1999), 1221-1231. doi: 10.1017/S0143385799141658.

[6]

M. Büger, Self-similarity of Julia sets of the composition of polynomials, Ergodic Theory Dynam. Systems, 17 (1997), 1289-1297. doi: 10.1017/S0143385797086458.

[7]

M. Büger, On the composition of polynomials of the form $z\sp 2+c\sb n$, Math. Ann., 310 (1998), 661-683.

[8]

R. Devaney, " An Introduction to Chaotic Dynamical Systems," Reprint of the second (1989) edition, Studies in Nonlinearity, Westview Press, Boulder, CO, 2003.

[9]

K. Falconer, "Techniques in Fractal Geometry," John Wiley & Sons, 1997.

[10]

J. E. Fornaess and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708. doi: 10.1017/S0143385700006428.

[11]

Z. Gong, W. Qiu and Y. Li, Connectedness of Julia sets for a quadratic random dynamical system, Ergodic Theory Dynam. Systems, 23 (2003), 1807-1815. doi: 10.1017/S0143385703000129.

[12]

Z. Gong and F. Ren, A random dynamical system formed by infinitely many functions, Journal of Fudan University, 35 (1996), 387-392.

[13]

A. Hinkkanen and G. J. Martin, The Dynamics of Semigroups of Rational Functions I, Proc. London Math. Soc. (3), 73 (1996), 358-384. doi: 10.1112/plms/s3-73.2.358.

[14]

A. Hinkkanen and G. J. Martin, Julia Sets of Rational Semigroups, Math. Z., 222 (1996), 161-169.

[15]

O. Lehto and K. I. Virtanen, "Quasiconformal Mappings in the Plane," Springer-Verlag, 1973.

[16]

J. Milnor, "Dynamics in One Complex Variable (Third Edition)," Annals of Mathematical Studies, 160, Princeton University Press, 2006.

[17]

S. B. Nadler, "Continuum Theory: An introduction," Marcel Dekker, 1992.

[18]

E. H. Spanier, "Algebraic Topology," Springer-Verlag, New York-Berlin, 1981.

[19]

R. Stankewitz, Completely invariant Julia sets of polynomial semigroups, Proc. Amer. Math. Soc., 127 (1999), 2889-2898. doi: 10.1090/S0002-9939-99-04857-1.

[20]

R. Stankewitz, Completely invariant sets of normality for rational semigroups, Complex Variables Theory Appl., 40 (2000), 199-210.

[21]

R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575. doi: 10.1090/S0002-9939-00-05313-2.

[22]

R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357-366.

[23]

R. Stankewitz and H. Sumi, Structure of Julia sets of polynomial semigroups with bounded finite postcritical set, Appl. Math. Comput., 187 (2007), 479-488. (Proceedings paper of a conference. This is not a full paper.) doi: 10.1016/j.amc.2006.08.148.

[24]

R. Stankewitz and H. Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups,, to appear in Trans. Amer. Math. Soc., (). 

[25]

N. Steinmetz, "Rational Iteration," de Gruyter Studies in Mathematics 16, Walter de Gruyter, 1993.

[26]

D. Steinsaltz, Random logistic maps and Lyapunov exponents, Indag. Mathem. N. S., 12 (2001), 557-584. doi: 10.1016/S0019-3577(01)80042-2.

[27]

H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.

[28]

H. Sumi, A correction to the proof of a lemma in 'Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products', Ergodic Theory Dynam. Systems, 21 (2001), 1275-1276. doi: 10.1017/S0143385701001602.

[29]

H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019. doi: 10.1088/0951-7715/13/4/302.

[30]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922. doi: 10.1017/S0143385705000532.

[31]

H. Sumi, On dynamics of hyperbolic rational semigroups, Journal of Mathematics of Kyoto University, 37 (1997), 717-733.

[32]

H. Sumi, Dimensions of Julia sets of expanding rational semigroups, Kodai Mathematical Journal, 28 (2005), 390-422. (See also arXiv:math/0405522.)

[33]

H. Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane, Applied Mathematics and Computation, 187 (2007), 489-500. (Proceedings paper of a conference. This is not a full paper.) doi: 10.1016/j.amc.2006.08.149.

[34]

H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity, RIMS Kokyuroku, 1494 (2006), 62-86. (Proceedings paper of a conference. This is not a full paper.)

[35]

H. Sumi, Interaction cohomology of forward or backward self-similar systems, Adv. Math., 222 (2009), 729-781. doi: 10.1016/j.aim.2009.04.007.

[36]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, (). 

[37]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups III: Classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles,, to appear in Ergodic Theory Dynam. Systems, (). 

[38]

H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, to appear in Proc. London Math. Soc., (). 

[39]

H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane,, survey article, (). 

[40]

H. Sumi, Cooperation principle in random complex dynamics and singular functions on the complex plane,, to appear in RIMS Kokyuroku. (Proceedings paper.), (). 

[41]

H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics, preprint (2010), arXiv:1008.3995.

[42]

, H. Sumi,, in preparation., (). 

[43]

H. Sumi and M. Urbański, The equilibrium states for semigroups of rational maps, Monatsh. Math., 156 (2009), 371-390. doi: 10.1007/s00605-008-0016-8.

[44]

H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups, Ergodic Theory Dynam. Systems, 30 (2010), 601-633. doi: 10.1017/S0143385709000297.

[45]

H. Sumi and M. Urbański, Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups,, to appear in Discrete and Continuous Dynamical Systems Ser. A, (). 

[46]

H. Sumi and M. Urbański, Bowen parameter and Hausdorff dimension for expanding rational semigroups, preprint (2009), arXiv:0911.3727.

[47]

Y. Sun and C-C. Yang, On the connectivity of the Julia set of a finitely generated rational semigroup, Proc. Amer. Math. Soc., 130 (2001), 49-52. doi: 10.1090/S0002-9939-01-06097-X.

[48]

W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Science Bulletin, 37 (1992), 969-971.

show all references

References:
[1]

L. V. Ahlfors, "Conformal Invariants: Topics in Geometric Function Theory," McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Dusseldorf-Johannesburg, 1973.

[2]

A. F. Beardon, Symmetries of Julia sets, Bull. London Math. Soc., 22 (1990), 576-582. doi: 10.1112/blms/22.6.576.

[3]

A. F. Beardon, "Iteration of Rational Functions," Graduate Text of Mathematics 132, Springer-Verlag, 1991.

[4]

R. Brück, Geometric properties of Julia sets of the composition of polynomials of the form $z^{2}+c_n$, Pacific J. Math., 198 (2001), 347-372. doi: 10.2140/pjm.2001.198.347.

[5]

R. Brück, M. Büger and S. Reitz, Random iterations of polynomials of the form $z^{2}+c_n$: Connectedness of Julia sets, Ergodic Theory Dynam. Systems, 19 (1999), 1221-1231. doi: 10.1017/S0143385799141658.

[6]

M. Büger, Self-similarity of Julia sets of the composition of polynomials, Ergodic Theory Dynam. Systems, 17 (1997), 1289-1297. doi: 10.1017/S0143385797086458.

[7]

M. Büger, On the composition of polynomials of the form $z\sp 2+c\sb n$, Math. Ann., 310 (1998), 661-683.

[8]

R. Devaney, " An Introduction to Chaotic Dynamical Systems," Reprint of the second (1989) edition, Studies in Nonlinearity, Westview Press, Boulder, CO, 2003.

[9]

K. Falconer, "Techniques in Fractal Geometry," John Wiley & Sons, 1997.

[10]

J. E. Fornaess and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708. doi: 10.1017/S0143385700006428.

[11]

Z. Gong, W. Qiu and Y. Li, Connectedness of Julia sets for a quadratic random dynamical system, Ergodic Theory Dynam. Systems, 23 (2003), 1807-1815. doi: 10.1017/S0143385703000129.

[12]

Z. Gong and F. Ren, A random dynamical system formed by infinitely many functions, Journal of Fudan University, 35 (1996), 387-392.

[13]

A. Hinkkanen and G. J. Martin, The Dynamics of Semigroups of Rational Functions I, Proc. London Math. Soc. (3), 73 (1996), 358-384. doi: 10.1112/plms/s3-73.2.358.

[14]

A. Hinkkanen and G. J. Martin, Julia Sets of Rational Semigroups, Math. Z., 222 (1996), 161-169.

[15]

O. Lehto and K. I. Virtanen, "Quasiconformal Mappings in the Plane," Springer-Verlag, 1973.

[16]

J. Milnor, "Dynamics in One Complex Variable (Third Edition)," Annals of Mathematical Studies, 160, Princeton University Press, 2006.

[17]

S. B. Nadler, "Continuum Theory: An introduction," Marcel Dekker, 1992.

[18]

E. H. Spanier, "Algebraic Topology," Springer-Verlag, New York-Berlin, 1981.

[19]

R. Stankewitz, Completely invariant Julia sets of polynomial semigroups, Proc. Amer. Math. Soc., 127 (1999), 2889-2898. doi: 10.1090/S0002-9939-99-04857-1.

[20]

R. Stankewitz, Completely invariant sets of normality for rational semigroups, Complex Variables Theory Appl., 40 (2000), 199-210.

[21]

R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575. doi: 10.1090/S0002-9939-00-05313-2.

[22]

R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357-366.

[23]

R. Stankewitz and H. Sumi, Structure of Julia sets of polynomial semigroups with bounded finite postcritical set, Appl. Math. Comput., 187 (2007), 479-488. (Proceedings paper of a conference. This is not a full paper.) doi: 10.1016/j.amc.2006.08.148.

[24]

R. Stankewitz and H. Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups,, to appear in Trans. Amer. Math. Soc., (). 

[25]

N. Steinmetz, "Rational Iteration," de Gruyter Studies in Mathematics 16, Walter de Gruyter, 1993.

[26]

D. Steinsaltz, Random logistic maps and Lyapunov exponents, Indag. Mathem. N. S., 12 (2001), 557-584. doi: 10.1016/S0019-3577(01)80042-2.

[27]

H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.

[28]

H. Sumi, A correction to the proof of a lemma in 'Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products', Ergodic Theory Dynam. Systems, 21 (2001), 1275-1276. doi: 10.1017/S0143385701001602.

[29]

H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019. doi: 10.1088/0951-7715/13/4/302.

[30]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922. doi: 10.1017/S0143385705000532.

[31]

H. Sumi, On dynamics of hyperbolic rational semigroups, Journal of Mathematics of Kyoto University, 37 (1997), 717-733.

[32]

H. Sumi, Dimensions of Julia sets of expanding rational semigroups, Kodai Mathematical Journal, 28 (2005), 390-422. (See also arXiv:math/0405522.)

[33]

H. Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane, Applied Mathematics and Computation, 187 (2007), 489-500. (Proceedings paper of a conference. This is not a full paper.) doi: 10.1016/j.amc.2006.08.149.

[34]

H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity, RIMS Kokyuroku, 1494 (2006), 62-86. (Proceedings paper of a conference. This is not a full paper.)

[35]

H. Sumi, Interaction cohomology of forward or backward self-similar systems, Adv. Math., 222 (2009), 729-781. doi: 10.1016/j.aim.2009.04.007.

[36]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, (). 

[37]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups III: Classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles,, to appear in Ergodic Theory Dynam. Systems, (). 

[38]

H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, to appear in Proc. London Math. Soc., (). 

[39]

H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane,, survey article, (). 

[40]

H. Sumi, Cooperation principle in random complex dynamics and singular functions on the complex plane,, to appear in RIMS Kokyuroku. (Proceedings paper.), (). 

[41]

H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics, preprint (2010), arXiv:1008.3995.

[42]

, H. Sumi,, in preparation., (). 

[43]

H. Sumi and M. Urbański, The equilibrium states for semigroups of rational maps, Monatsh. Math., 156 (2009), 371-390. doi: 10.1007/s00605-008-0016-8.

[44]

H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups, Ergodic Theory Dynam. Systems, 30 (2010), 601-633. doi: 10.1017/S0143385709000297.

[45]

H. Sumi and M. Urbański, Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups,, to appear in Discrete and Continuous Dynamical Systems Ser. A, (). 

[46]

H. Sumi and M. Urbański, Bowen parameter and Hausdorff dimension for expanding rational semigroups, preprint (2009), arXiv:0911.3727.

[47]

Y. Sun and C-C. Yang, On the connectivity of the Julia set of a finitely generated rational semigroup, Proc. Amer. Math. Soc., 130 (2001), 49-52. doi: 10.1090/S0002-9939-01-06097-X.

[48]

W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Science Bulletin, 37 (1992), 969-971.

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