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 - A, 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.

[2]

Walter Bergweiler, The role of the Ahlfors five islands theorem in complex dynamics,, Conform. Geom. Dyn., 4 (2000), 22. doi: 10.1090/S1088-4173-00-00057-6.

[3]

François Berteloot and Julien Duval, Une démonstration directe de la densité des cycles répulsifs dans l'ensemble de Julia,, in, 188 (2000), 221.

[4]

F. W. Gehring and G. J. Martin, Iteration theory and inequalities for Kleinian groups,, Bull. Amer. Math. Soc. (N.S.), 21 (1989), 57.

[5]

F. W. Gehring and G. J. Martin, Commutators, collars and the geometry of Möbius groups,, J. Anal. Math., 63 (1994), 175. doi: 10.1007/BF03008423.

[6]

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

[7]

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.

[8]

Hartje Kriete and Hiroki Sumi, Semihyperbolic transcendental semigroups,, J. Math. Kyoto Univ., 40 (2000), 205.

[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, (2000).

[10]

Wilhelm Schwick, Normality criteria for families of meromorphic functions,, J. Analyse Math., 52 (1989), 241. doi: 10.1007/BF02820480.

[11]

Wilhelm Schwick, Repelling periodic points in the Julia set,, Bull. London Math. Soc., 29 (1989), 314. doi: 10.1112/S0024609396007035.

[12]

Rich 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.

[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. doi: 10.1080/10236190903203929.

[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. doi: 10.1090/S0002-9947-2011-05199-8.

[15]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets,, Discrete Contin. Dyn. Syst., 29 (2011), 1205. doi: 10.3934/dcds.2011.29.1205.

[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. doi: 10.1017/S0143385709000923.

[17]

H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Math. J., 28 (2005), 390. doi: 10.2996/kmj/1123767019.

[18]

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

[19]

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

[20]

H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, Proc. London Math. Soc. (3), 102 (2011), 50. doi: 10.1112/plms/pdq013.

[21]

H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane,, Sūgaku, 61 (2009), 133.

[22]

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.

[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.

[24]

Hiroki Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane,, Appl. Math. Comput., 187 (2007), 489. doi: 10.1016/j.amc.2006.08.149.

[25]

Hiroki Sumi, Dynamics of postcritically bounded polynomial semigroups,, preprint, ().

[26]

Lawrence Zalcman, A heuristic principle in complex function theory,, Amer. Math. Monthly, 82 (1975), 813. doi: 10.2307/2319796.

[27]

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

show all references

References:
[1]

Walter Bergweiler, A new proof of the Ahlfors five islands theorem,, J. Anal. Math., 76 (1998), 337.

[2]

Walter Bergweiler, The role of the Ahlfors five islands theorem in complex dynamics,, Conform. Geom. Dyn., 4 (2000), 22. doi: 10.1090/S1088-4173-00-00057-6.

[3]

François Berteloot and Julien Duval, Une démonstration directe de la densité des cycles répulsifs dans l'ensemble de Julia,, in, 188 (2000), 221.

[4]

F. W. Gehring and G. J. Martin, Iteration theory and inequalities for Kleinian groups,, Bull. Amer. Math. Soc. (N.S.), 21 (1989), 57.

[5]

F. W. Gehring and G. J. Martin, Commutators, collars and the geometry of Möbius groups,, J. Anal. Math., 63 (1994), 175. doi: 10.1007/BF03008423.

[6]

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

[7]

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.

[8]

Hartje Kriete and Hiroki Sumi, Semihyperbolic transcendental semigroups,, J. Math. Kyoto Univ., 40 (2000), 205.

[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, (2000).

[10]

Wilhelm Schwick, Normality criteria for families of meromorphic functions,, J. Analyse Math., 52 (1989), 241. doi: 10.1007/BF02820480.

[11]

Wilhelm Schwick, Repelling periodic points in the Julia set,, Bull. London Math. Soc., 29 (1989), 314. doi: 10.1112/S0024609396007035.

[12]

Rich 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.

[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. doi: 10.1080/10236190903203929.

[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. doi: 10.1090/S0002-9947-2011-05199-8.

[15]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets,, Discrete Contin. Dyn. Syst., 29 (2011), 1205. doi: 10.3934/dcds.2011.29.1205.

[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. doi: 10.1017/S0143385709000923.

[17]

H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Math. J., 28 (2005), 390. doi: 10.2996/kmj/1123767019.

[18]

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

[19]

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

[20]

H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, Proc. London Math. Soc. (3), 102 (2011), 50. doi: 10.1112/plms/pdq013.

[21]

H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane,, Sūgaku, 61 (2009), 133.

[22]

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.

[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.

[24]

Hiroki Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane,, Appl. Math. Comput., 187 (2007), 489. doi: 10.1016/j.amc.2006.08.149.

[25]

Hiroki Sumi, Dynamics of postcritically bounded polynomial semigroups,, preprint, ().

[26]

Lawrence Zalcman, A heuristic principle in complex function theory,, Amer. Math. Monthly, 82 (1975), 813. doi: 10.2307/2319796.

[27]

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

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