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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 & Continuous Dynamical Systems - A, 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, (1973).   Google Scholar [2] A. F. Beardon, Symmetries of Julia sets,, Bull. London Math. Soc., 22 (1990), 576.  doi: 10.1112/blms/22.6.576.  Google Scholar [3] A. F. Beardon, "Iteration of Rational Functions,", Graduate Text of Mathematics \textbf{132}, 132 (1991).   Google Scholar [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.  doi: 10.2140/pjm.2001.198.347.  Google Scholar [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.  doi: 10.1017/S0143385799141658.  Google Scholar [6] M. Büger, Self-similarity of Julia sets of the composition of polynomials,, Ergodic Theory Dynam. Systems, 17 (1997), 1289.  doi: 10.1017/S0143385797086458.  Google Scholar [7] M. Büger, On the composition of polynomials of the form $z\sp 2+c\sb n$,, Math. Ann., 310 (1998), 661.   Google Scholar [8] R. Devaney, " An Introduction to Chaotic Dynamical Systems,", Reprint of the second (1989) edition, (1989).   Google Scholar [9] K. Falconer, "Techniques in Fractal Geometry,", John Wiley & Sons, (1997).   Google Scholar [10] J. E. Fornaess and N. Sibony, Random iterations of rational functions,, Ergodic Theory Dynam. Systems, 11 (1991), 687.  doi: 10.1017/S0143385700006428.  Google Scholar [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.  doi: 10.1017/S0143385703000129.  Google Scholar [12] Z. Gong and F. Ren, A random dynamical system formed by infinitely many functions,, Journal of Fudan University, 35 (1996), 387.   Google Scholar [13] A. Hinkkanen and G. J. Martin, The Dynamics of Semigroups of Rational Functions I,, Proc. London Math. Soc. (3), 73 (1996), 358.  doi: 10.1112/plms/s3-73.2.358.  Google Scholar [14] A. Hinkkanen and G. J. Martin, Julia Sets of Rational Semigroups,, Math. Z., 222 (1996), 161.   Google Scholar [15] O. Lehto and K. I. Virtanen, "Quasiconformal Mappings in the Plane,", Springer-Verlag, (1973).   Google Scholar [16] J. Milnor, "Dynamics in One Complex Variable (Third Edition),", Annals of Mathematical Studies, 160 (2006).   Google Scholar [17] S. B. Nadler, "Continuum Theory: An introduction,", Marcel Dekker, (1992).   Google Scholar [18] E. H. Spanier, "Algebraic Topology,", Springer-Verlag, (1981).   Google Scholar [19] R. Stankewitz, Completely invariant Julia sets of polynomial semigroups,, Proc. Amer. Math. Soc., 127 (1999), 2889.  doi: 10.1090/S0002-9939-99-04857-1.  Google Scholar [20] R. Stankewitz, Completely invariant sets of normality for rational semigroups,, Complex Variables Theory Appl., 40 (2000), 199.   Google Scholar [21] R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's,, Proc. Amer. Math. Soc., 128 (2000), 2569.  doi: 10.1090/S0002-9939-00-05313-2.  Google Scholar [22] R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups,, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357.   Google Scholar [23] R. Stankewitz and H. Sumi, Structure of Julia sets of polynomial semigroups with bounded finite postcritical set,, Appl. Math. Comput., 187 (2007), 479.  doi: 10.1016/j.amc.2006.08.148.  Google Scholar [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., ().   Google Scholar [25] N. Steinmetz, "Rational Iteration,", de Gruyter Studies in Mathematics \textbf{16}, 16 (1993).   Google Scholar [26] D. Steinsaltz, Random logistic maps and Lyapunov exponents,, Indag. Mathem. N. S., 12 (2001), 557.  doi: 10.1016/S0019-3577(01)80042-2.  Google Scholar [27] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products,, Ergodic Theory Dynam. Systems, 21 (2001), 563.   Google Scholar [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.  doi: 10.1017/S0143385701001602.  Google Scholar [29] H. Sumi, Skew product maps related to finitely generated rational semigroups,, Nonlinearity, 13 (2000), 995.  doi: 10.1088/0951-7715/13/4/302.  Google Scholar [30] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups,, Ergodic Theory Dynam. Systems, 26 (2006), 893.  doi: 10.1017/S0143385705000532.  Google Scholar [31] H. Sumi, On dynamics of hyperbolic rational semigroups,, Journal of Mathematics of Kyoto University, 37 (1997), 717.   Google Scholar [32] H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Mathematical Journal, 28 (2005), 390.   Google Scholar [33] H. Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane,, Applied Mathematics and Computation, 187 (2007), 489.  doi: 10.1016/j.amc.2006.08.149.  Google Scholar [34] H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity,, RIMS Kokyuroku, 1494 (2006), 62.   Google Scholar [35] H. Sumi, Interaction cohomology of forward or backward self-similar systems,, Adv. Math., 222 (2009), 729.  doi: 10.1016/j.aim.2009.04.007.  Google Scholar [36] H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, ().   Google Scholar [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, ().   Google Scholar [38] H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, to appear in Proc. London Math. Soc., ().   Google Scholar [39] H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane,, survey article, ().   Google Scholar [40] H. Sumi, Cooperation principle in random complex dynamics and singular functions on the complex plane,, to appear in RIMS Kokyuroku. (Proceedings paper.), ().   Google Scholar [41] H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics,, preprint (2010), (2010).   Google Scholar [42] , H. Sumi,, in preparation., ().   Google Scholar [43] H. Sumi and M. Urbański, The equilibrium states for semigroups of rational maps,, Monatsh. Math., 156 (2009), 371.  doi: 10.1007/s00605-008-0016-8.  Google Scholar [44] H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups,, Ergodic Theory Dynam. Systems, 30 (2010), 601.  doi: 10.1017/S0143385709000297.  Google Scholar [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, ().   Google Scholar [46] H. Sumi and M. Urbański, Bowen parameter and Hausdorff dimension for expanding rational semigroups,, preprint (2009), (2009).   Google Scholar [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.  doi: 10.1090/S0002-9939-01-06097-X.  Google Scholar [48] W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions,, Chinese Science Bulletin, 37 (1992), 969.   Google Scholar

show all references

##### References:
 [1] L. V. Ahlfors, "Conformal Invariants: Topics in Geometric Function Theory,", McGraw-Hill Series in Higher Mathematics, (1973).   Google Scholar [2] A. F. Beardon, Symmetries of Julia sets,, Bull. London Math. Soc., 22 (1990), 576.  doi: 10.1112/blms/22.6.576.  Google Scholar [3] A. F. Beardon, "Iteration of Rational Functions,", Graduate Text of Mathematics \textbf{132}, 132 (1991).   Google Scholar [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.  doi: 10.2140/pjm.2001.198.347.  Google Scholar [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.  doi: 10.1017/S0143385799141658.  Google Scholar [6] M. Büger, Self-similarity of Julia sets of the composition of polynomials,, Ergodic Theory Dynam. Systems, 17 (1997), 1289.  doi: 10.1017/S0143385797086458.  Google Scholar [7] M. Büger, On the composition of polynomials of the form $z\sp 2+c\sb n$,, Math. Ann., 310 (1998), 661.   Google Scholar [8] R. Devaney, " An Introduction to Chaotic Dynamical Systems,", Reprint of the second (1989) edition, (1989).   Google Scholar [9] K. Falconer, "Techniques in Fractal Geometry,", John Wiley & Sons, (1997).   Google Scholar [10] J. E. Fornaess and N. Sibony, Random iterations of rational functions,, Ergodic Theory Dynam. Systems, 11 (1991), 687.  doi: 10.1017/S0143385700006428.  Google Scholar [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.  doi: 10.1017/S0143385703000129.  Google Scholar [12] Z. Gong and F. Ren, A random dynamical system formed by infinitely many functions,, Journal of Fudan University, 35 (1996), 387.   Google Scholar [13] A. Hinkkanen and G. J. Martin, The Dynamics of Semigroups of Rational Functions I,, Proc. London Math. Soc. (3), 73 (1996), 358.  doi: 10.1112/plms/s3-73.2.358.  Google Scholar [14] A. Hinkkanen and G. J. Martin, Julia Sets of Rational Semigroups,, Math. Z., 222 (1996), 161.   Google Scholar [15] O. Lehto and K. I. Virtanen, "Quasiconformal Mappings in the Plane,", Springer-Verlag, (1973).   Google Scholar [16] J. Milnor, "Dynamics in One Complex Variable (Third Edition),", Annals of Mathematical Studies, 160 (2006).   Google Scholar [17] S. B. Nadler, "Continuum Theory: An introduction,", Marcel Dekker, (1992).   Google Scholar [18] E. H. Spanier, "Algebraic Topology,", Springer-Verlag, (1981).   Google Scholar [19] R. Stankewitz, Completely invariant Julia sets of polynomial semigroups,, Proc. Amer. Math. Soc., 127 (1999), 2889.  doi: 10.1090/S0002-9939-99-04857-1.  Google Scholar [20] R. Stankewitz, Completely invariant sets of normality for rational semigroups,, Complex Variables Theory Appl., 40 (2000), 199.   Google Scholar [21] R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's,, Proc. Amer. Math. Soc., 128 (2000), 2569.  doi: 10.1090/S0002-9939-00-05313-2.  Google Scholar [22] R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups,, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357.   Google Scholar [23] R. Stankewitz and H. Sumi, Structure of Julia sets of polynomial semigroups with bounded finite postcritical set,, Appl. Math. Comput., 187 (2007), 479.  doi: 10.1016/j.amc.2006.08.148.  Google Scholar [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., ().   Google Scholar [25] N. Steinmetz, "Rational Iteration,", de Gruyter Studies in Mathematics \textbf{16}, 16 (1993).   Google Scholar [26] D. Steinsaltz, Random logistic maps and Lyapunov exponents,, Indag. Mathem. N. S., 12 (2001), 557.  doi: 10.1016/S0019-3577(01)80042-2.  Google Scholar [27] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products,, Ergodic Theory Dynam. Systems, 21 (2001), 563.   Google Scholar [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.  doi: 10.1017/S0143385701001602.  Google Scholar [29] H. Sumi, Skew product maps related to finitely generated rational semigroups,, Nonlinearity, 13 (2000), 995.  doi: 10.1088/0951-7715/13/4/302.  Google Scholar [30] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups,, Ergodic Theory Dynam. Systems, 26 (2006), 893.  doi: 10.1017/S0143385705000532.  Google Scholar [31] H. Sumi, On dynamics of hyperbolic rational semigroups,, Journal of Mathematics of Kyoto University, 37 (1997), 717.   Google Scholar [32] H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Mathematical Journal, 28 (2005), 390.   Google Scholar [33] H. Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane,, Applied Mathematics and Computation, 187 (2007), 489.  doi: 10.1016/j.amc.2006.08.149.  Google Scholar [34] H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity,, RIMS Kokyuroku, 1494 (2006), 62.   Google Scholar [35] H. Sumi, Interaction cohomology of forward or backward self-similar systems,, Adv. Math., 222 (2009), 729.  doi: 10.1016/j.aim.2009.04.007.  Google Scholar [36] H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, ().   Google Scholar [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, ().   Google Scholar [38] H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, to appear in Proc. London Math. Soc., ().   Google Scholar [39] H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane,, survey article, ().   Google Scholar [40] H. Sumi, Cooperation principle in random complex dynamics and singular functions on the complex plane,, to appear in RIMS Kokyuroku. (Proceedings paper.), ().   Google Scholar [41] H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics,, preprint (2010), (2010).   Google Scholar [42] , H. Sumi,, in preparation., ().   Google Scholar [43] H. Sumi and M. Urbański, The equilibrium states for semigroups of rational maps,, Monatsh. Math., 156 (2009), 371.  doi: 10.1007/s00605-008-0016-8.  Google Scholar [44] H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups,, Ergodic Theory Dynam. Systems, 30 (2010), 601.  doi: 10.1017/S0143385709000297.  Google Scholar [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, ().   Google Scholar [46] H. Sumi and M. Urbański, Bowen parameter and Hausdorff dimension for expanding rational semigroups,, preprint (2009), (2009).   Google Scholar [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.  doi: 10.1090/S0002-9939-01-06097-X.  Google Scholar [48] W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions,, Chinese Science Bulletin, 37 (1992), 969.   Google Scholar
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