Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups
HTML articles powered by AMS MathViewer
- by Rich Stankewitz and Hiroki Sumi PDF
- Trans. Amer. Math. Soc. 363 (2011), 5293-5319 Request permission
Abstract:
We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected. In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of $G.$ Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps $g \in G$ are distributed within the Julia set of the entire semigroup $G$. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition.References
- Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
- Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
- David A. Boyd, The immediate basin of attraction of infinity for polynomial semigroups of finite type, J. London Math. Soc. (2) 69 (2004), no. 1, 201–213. MR 2025336, DOI 10.1112/S0024610703004691
- Rainer Brück, Geometric properties of Julia sets of the composition of polynomials of the form $z^2+c_n$, Pacific J. Math. 198 (2001), no. 2, 347–372. MR 1835513, DOI 10.2140/pjm.2001.198.347
- Rainer Brück, Matthias Büger, and Stefan Reitz, Random iterations of polynomials of the form $z^2+c_n$: connectedness of Julia sets, Ergodic Theory Dynam. Systems 19 (1999), no. 5, 1221–1231. MR 1721617, DOI 10.1017/S0143385799141658
- Matthias Büger, Self-similarity of Julia sets of the composition of polynomials, Ergodic Theory Dynam. Systems 17 (1997), no. 6, 1289–1297. MR 1488318, DOI 10.1017/S0143385797086458
- Matthias Büger, On the composition of polynomials of the form $z^2+c_n$, Math. Ann. 310 (1998), no. 4, 661–683. MR 1619744, DOI 10.1007/s002080050165
- Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
- John Erik Fornæss and Nessim Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 687–708. MR 1145616, DOI 10.1017/S0143385700006428
- Zhimin Gong and Fuyao Ren, A random dynamical system formed by infinitely many functions, J. Fudan Univ. Nat. Sci. 35 (1996), no. 4, 387–392 (English, with English and Chinese summaries). MR 1435167
- A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions. I, Proc. London Math. Soc. (3) 73 (1996), no. 2, 358–384. MR 1397693, DOI 10.1112/plms/s3-73.2.358
- Sam B. Nadler Jr., Continuum theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR 1192552
- Rich Stankewitz. Completely invariant Julia sets of rational semigroups. Ph.D. thesis, University of Illinois, 1998.
- Rich Stankewitz, Completely invariant Julia sets of polynomial semigroups, Proc. Amer. Math. Soc. 127 (1999), no. 10, 2889–2898. MR 1600149, DOI 10.1090/S0002-9939-99-04857-1
- Rich Stankewitz, Completely invariant sets of normality for rational semigroups, Complex Variables Theory Appl. 40 (2000), no. 3, 199–210. MR 1753707, DOI 10.1080/17476930008815219
- Rich Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, J. Difference Equ. Appl. 16 (2010), no. 5-6, 763–771. MR 2675604, DOI 10.1080/10236190903203929
- Rich Stankewitz, W. Conatser, T. Butz, B. Dean, Y. Li, and K. Hart. JULIA 2.0 Fractal Drawing Program. http://rstankewitz.iweb.bsu.edu/JuliaHelp2.0/Julia.html.
- Rich Stankewitz, Toshiyuki Sugawa, and Hiroki Sumi, Some counterexamples in dynamics of rational semigroups, Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 2, 357–366. MR 2097238
- Rich Stankewitz and Hiroki Sumi, Structure of Julia sets of polynomial semigroups with bounded finite postcritical set, Appl. Math. Comput. 187 (2007), no. 1, 479–488. MR 2323604, DOI 10.1016/j.amc.2006.08.148
- Hiroki Sumi, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups, Kodai Math. J. 21 (1998), no. 1, 10–28. MR 1625124, DOI 10.2996/kmj/1138043831
- Hiroki Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity 13 (2000), no. 4, 995–1019. MR 1767945, DOI 10.1088/0951-7715/13/4/302
- Hiroki Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 563–603. MR 1827119, DOI 10.1017/S0143385701001286
- Hiroki Sumi, Dimensions of Julia sets of expanding rational semigroups, Kodai Math. J. 28 (2005), no. 2, 390–422. MR 2153926, DOI 10.2996/kmj/1123767019
- Hiroki Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems 26 (2006), no. 3, 893–922. MR 2237476, DOI 10.1017/S0143385705000532
- Hiroki Sumi. Interaction cohomology of forward or backward self-similar systems. Adv. Math., 222:729–781, 1009.
- Hiroki Sumi. Dynamics of polynomial semigroups with bounded postcritical set in the plane. RIMS Kokyuroku, 1447:198–215, 2005. (Proceedings paper.)
- Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups and interaction cohomology. RIMS Kokyuroku, 1447:227–238, 2005. (Proceedings paper.)
- Hiroki Sumi, Random dynamics of polynomials and devil’s-staircase-like functions in the complex plane, Appl. Math. Comput. 187 (2007), no. 1, 489–500. MR 2323605, DOI 10.1016/j.amc.2006.08.149
- Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets. Discrete and Continuous Dynamical Systems Series A, Vol. 29, No. 3, 2011, 1205–1244.
- Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets. Preprint 2008, http://arxiv.org/abs/1007.0613.
- Hiroki 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), no. 6, 1869–1902. MR 2736899, DOI 10.1017/S0143385709000923
- Hiroki Sumi, Random complex dynamics and semigroups of holomorphic maps, Proc. Lond. Math. Soc. (3) 102 (2011), no. 1, 50–112. MR 2747724, DOI 10.1112/plms/pdq013
- Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups. Preprint 2006, http://arxiv.org/abs/math.DS/0703591.
- Random complex dynamics and devil’s coliseums. Preprint 2011, http://arxiv.org/abs/ 1104.3640.
- Hiroki Sumi and Mariusz Urbański, The equilibrium states for semigroups of rational maps, Monatsh. Math. 156 (2009), no. 4, 371–390. MR 2486604, DOI 10.1007/s00605-008-0016-8
- Hiroki Sumi and Mariusz Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups, Ergodic Theory Dynam. Systems 30 (2010), no. 2, 601–633. MR 2599895, DOI 10.1017/S0143385709000297
- Hiroki Sumi and Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete and Continuous Dynamical Systems Ser. A., Vol. 30, No. 1, 2011, 313–363.
- Yeshun Sun and Chung-Chun Yang, On the connectivity of the Julia set of a finitely generated rational semigroup, Proc. Amer. Math. Soc. 130 (2002), no. 1, 49–52. MR 1855618, DOI 10.1090/S0002-9939-01-06097-X
- W. Zhou and F. Ren. The Julia sets of the random iteration of rational functions. Chinese Sci. Bulletin, 37(12):969–971, 1992.
Additional Information
- Rich Stankewitz
- Affiliation: Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47306
- Email: rstankewitz@bsu.edu
- Hiroki Sumi
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan
- MR Author ID: 622791
- Email: sumi@math.sci.osaka-u.ac.jp
- Received by editor(s): May 14, 2007
- Received by editor(s) in revised form: August 13, 2009
- Published electronically: May 20, 2011
- Additional Notes: The first author was partially supported by the BSU Lilly V grant. He would also like to thank Osaka University for their hospitality during his stay there while this work was begun.
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 5293-5319
- MSC (2010): Primary 37F10, 37F50, 30D05
- DOI: https://doi.org/10.1090/S0002-9947-2011-05199-8
- MathSciNet review: 2813416