# American Institute of Mathematical Sciences

July  2012, 32(7): 2583-2589. doi: 10.3934/dcds.2012.32.2583

## Density of repelling fixed points in the Julia set of a rational or entire semigroup, II

 1 Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, United States

Received  November 2009 Revised  March 2010 Published  March 2012

In [13] there is a survey of several methods of proof that the Julia set of a rational or entire function is the closure of the repelling cycles, along with a discussion of which of those methods can and cannot be extended to the case of semigroups. In particular that paper presents an elementary proof based on the ideas of [11] that the Julia set of either a non-elementary rational or entire semigroup is the closure of the set of repelling fixed points. This paper serves as a brief follow up to [13] by showing that the ideas of [3] can also be used to provide an elementary proof for the semigroup case. It also touches upon some key differences between the dynamics of iteration and the dynamics of semigroups.
Citation: Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583
##### References:
 [1] Walter Bergweiler, A new proof of the Ahlfors five islands theorem, J. Anal. Math., 76 (1998), 337-347.  Google Scholar [2] Walter Bergweiler, The role of the Ahlfors five islands theorem in complex dynamics, Conform. Geom. Dyn., 4 (2000), 22-34 (electronic). doi: 10.1090/S1088-4173-00-00057-6.  Google Scholar [3] François Berteloot and Julien Duval, Une démonstration directe de la densité des cycles répulsifs dans l'ensemble de Julia, in "Complex Analysis and Geometry" (Paris, 1997), Progr. Math., 188, Birkhäuser, Basel, (2000), 221-222.  Google Scholar [4] F. W. Gehring and G. J. Martin, Iteration theory and inequalities for Kleinian groups, Bull. Amer. Math. Soc. (N.S.), 21 (1989), 57-63.  Google Scholar [5] F. W. Gehring and G. J. Martin, Commutators, collars and the geometry of Möbius groups, J. Anal. Math., 63 (1994), 175-219. doi: 10.1007/BF03008423.  Google Scholar [6] Z. Gong and F. Ren, A random dynamical system formed by infinitely many functions, Journal of Fudan University Nat. Sci., 35 (1996), 387-392.  Google Scholar [7] 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.  Google Scholar [8] Hartje Kriete and Hiroki Sumi, Semihyperbolic transcendental semigroups, J. Math. Kyoto Univ., 40 (2000), 205-216.  Google Scholar [9] S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, "Holomorphic Dynamics," Translated from the 1995 Japanese original and revised by the authors, Cambridge Studies in Advanced Mathematics, 66, Cambridge University Press, Cambridge, 2000.  Google Scholar [10] Wilhelm Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math., 52 (1989), 241-289. doi: 10.1007/BF02820480.  Google Scholar [11] Wilhelm Schwick, Repelling periodic points in the Julia set, Bull. London Math. Soc., 29 (1989), 314-316. doi: 10.1112/S0024609396007035.  Google Scholar [12] Rich 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.  Google Scholar [13] Rich Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, J. Difference Equ. Appl., 16 (2010), 763-771. doi: 10.1080/10236190903203929.  Google Scholar [14] R. Stankewitz and H. Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups, Trans. Amer. Math. Soc., 363 (2011), 5293-5319. doi: 10.1090/S0002-9947-2011-05199-8.  Google Scholar [15] H. Sumi, Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets, Discrete Contin. Dyn. Syst., 29 (2011), 1205-1244. doi: 10.3934/dcds.2011.29.1205.  Google Scholar [16] 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, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902. doi: 10.1017/S0143385709000923.  Google Scholar [17] H. Sumi, Dimensions of Julia sets of expanding rational semigroups, Kodai Math. J., 28 (2005), 390-422. doi: 10.2996/kmj/1123767019.  Google Scholar [18] 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.  Google Scholar [19] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922. doi: 10.1017/S0143385705000532.  Google Scholar [20] H. Sumi, Random complex dynamics and semigroups of holomorphic maps, Proc. London Math. Soc. (3), 102 (2011), 50-112. doi: 10.1112/plms/pdq013.  Google Scholar [21] H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane, Sūgaku, 61 (2009), 133-161.  Google Scholar [22] 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.  Google Scholar [23] H. Sumi and M. Urbański, Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups, Discrete and Continuous Dynamical Systems Ser. A, 32 (2012), 2591-2606. Google Scholar [24] Hiroki Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane, Appl. Math. Comput., 187 (2007), 489-500. doi: 10.1016/j.amc.2006.08.149.  Google Scholar [25] Hiroki Sumi, Dynamics of postcritically bounded polynomial semigroups,, preprint, ().   Google Scholar [26] Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly, 82 (1975), 813-817. doi: 10.2307/2319796.  Google Scholar [27] W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Bulletin, 37 (1992), 969-971. Google Scholar

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##### References:
 [1] Walter Bergweiler, A new proof of the Ahlfors five islands theorem, J. Anal. Math., 76 (1998), 337-347.  Google Scholar [2] Walter Bergweiler, The role of the Ahlfors five islands theorem in complex dynamics, Conform. Geom. Dyn., 4 (2000), 22-34 (electronic). doi: 10.1090/S1088-4173-00-00057-6.  Google Scholar [3] François Berteloot and Julien Duval, Une démonstration directe de la densité des cycles répulsifs dans l'ensemble de Julia, in "Complex Analysis and Geometry" (Paris, 1997), Progr. Math., 188, Birkhäuser, Basel, (2000), 221-222.  Google Scholar [4] F. W. Gehring and G. J. Martin, Iteration theory and inequalities for Kleinian groups, Bull. Amer. Math. Soc. (N.S.), 21 (1989), 57-63.  Google Scholar [5] F. W. Gehring and G. J. Martin, Commutators, collars and the geometry of Möbius groups, J. Anal. Math., 63 (1994), 175-219. doi: 10.1007/BF03008423.  Google Scholar [6] Z. Gong and F. Ren, A random dynamical system formed by infinitely many functions, Journal of Fudan University Nat. Sci., 35 (1996), 387-392.  Google Scholar [7] 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.  Google Scholar [8] Hartje Kriete and Hiroki Sumi, Semihyperbolic transcendental semigroups, J. Math. Kyoto Univ., 40 (2000), 205-216.  Google Scholar [9] S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, "Holomorphic Dynamics," Translated from the 1995 Japanese original and revised by the authors, Cambridge Studies in Advanced Mathematics, 66, Cambridge University Press, Cambridge, 2000.  Google Scholar [10] Wilhelm Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math., 52 (1989), 241-289. doi: 10.1007/BF02820480.  Google Scholar [11] Wilhelm Schwick, Repelling periodic points in the Julia set, Bull. London Math. Soc., 29 (1989), 314-316. doi: 10.1112/S0024609396007035.  Google Scholar [12] Rich 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.  Google Scholar [13] Rich Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, J. Difference Equ. Appl., 16 (2010), 763-771. doi: 10.1080/10236190903203929.  Google Scholar [14] R. Stankewitz and H. Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups, Trans. Amer. Math. Soc., 363 (2011), 5293-5319. doi: 10.1090/S0002-9947-2011-05199-8.  Google Scholar [15] H. Sumi, Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets, Discrete Contin. Dyn. Syst., 29 (2011), 1205-1244. doi: 10.3934/dcds.2011.29.1205.  Google Scholar [16] 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, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902. doi: 10.1017/S0143385709000923.  Google Scholar [17] H. Sumi, Dimensions of Julia sets of expanding rational semigroups, Kodai Math. J., 28 (2005), 390-422. doi: 10.2996/kmj/1123767019.  Google Scholar [18] 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.  Google Scholar [19] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922. doi: 10.1017/S0143385705000532.  Google Scholar [20] H. Sumi, Random complex dynamics and semigroups of holomorphic maps, Proc. London Math. Soc. (3), 102 (2011), 50-112. doi: 10.1112/plms/pdq013.  Google Scholar [21] H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane, Sūgaku, 61 (2009), 133-161.  Google Scholar [22] 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.  Google Scholar [23] H. Sumi and M. Urbański, Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups, Discrete and Continuous Dynamical Systems Ser. A, 32 (2012), 2591-2606. Google Scholar [24] Hiroki Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane, Appl. Math. Comput., 187 (2007), 489-500. doi: 10.1016/j.amc.2006.08.149.  Google Scholar [25] Hiroki Sumi, Dynamics of postcritically bounded polynomial semigroups,, preprint, ().   Google Scholar [26] Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly, 82 (1975), 813-817. doi: 10.2307/2319796.  Google Scholar [27] W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Bulletin, 37 (1992), 969-971. Google Scholar
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