DYNAMICS OF TRANSCENDENTAL ENTIRE FUNCTIONS WITH SIEGEL DISKS AND ITS APPLICATIONS

• Published : 2011.07.31

Abstract

We study the dynamics of transcendental entire functions with Siegel disks whose singular values are just two points. One of the two singular values is not only a superattracting fixed point with multiplicity more than two but also an asymptotic value. Another one is a critical value with free dynamics under iterations. We prove that if the multiplicity of the superattracting fixed point is large enough, then the restriction of the transcendental entire function near the Siegel point is a quadratic-like map. Therefore the Siegel disk and its boundary correspond to those of some quadratic polynomial at the level of quasiconformality. As its applications, the logarithmic lift of the above transcendental entire function has a wandering domain whose shape looks like a Siegel disk of a quadratic polynomial.

References

1. L. V. Ahlfors, Lectures on Quasiconformal Mappings, Second edition, With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. University Lecture Series, 38. American Mathematical Society, Providence, RI, 2006.
2. A. Avila, X. Buff and A. Cheritat, Siegel disks with smooth boundaries, Acta Math. 193 (2004), no. 1, 1-30. https://doi.org/10.1007/BF02392549
3. A. F. Beardon, Iteration of Rational Functions, Complex analytic dynamical systems, Graduate Texts in Mathematics, 132. Springer-Verlag, New York, 1991.
4. R. Berenguel and N. Fagella, An entire transcendental family with a peersistent Siegel disc, http://www.gsd.uab.es/personal/nfagella.
5. P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 85-141. https://doi.org/10.1090/S0273-0979-1984-15240-6
6. X. Buff and A. Cheritat, Quadratic Julia sets with positive area, http://www.picard.ups-tlse.fr/buff/Preprints/Preprints.html.
7. X. Buff and C. Henriksen, Scaling ratios and triangles in Siegel disks, Math. Res. Lett. 6 (1999), no. 3-4, 293-305. https://doi.org/10.4310/MRL.1999.v6.n3.a4
8. L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, 1993.
9. A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Ecole Norm. Sup. (4) 18 (1985), no. 2, 287-343. https://doi.org/10.24033/asens.1491
10. N. Fagella and L. Geyer, Surgery on Herman rings of the complex standard family, Ergodic Theory Dynam. Systems 23 (2003), no. 2, 493-508.
11. N. Fagella and C. Henriksen, The Teichmuller space of an entire function, Complex dynamics, 297-330, A K Peters, Wellesley, MA, 2009.
12. A. Fletcher and V. Markovic, Quasiconformal Maps and Teichmuller Theory, Oxford Graduate Texts in Mathematics 11, Oxford University Press, Oxford, 2007.
13. L. Geyer, Siegel discs, Herman rings and the Arnold family, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3661-3683. https://doi.org/10.1090/S0002-9947-01-02662-9
14. J. Graczyk and P. Jones, Dimension of the boundary of quasiconformal Siegel disks, Invent. Math. 148 (2002), no. 3, 465-493. https://doi.org/10.1007/s002220100198
15. K. Katagata, Some cubic Blaschke products and quadratic rational functions with Siegel disks, Int. J. Contemp. Math. Sci. 2 (2007), no. 30, 1455-1470. https://doi.org/10.12988/ijcms.2007.07153
16. K. Katagata, Dynamics of rational functions and rational semigroups on the Riemann sphere, Thesis, Shimane University, 2008.
17. K. Katagata, Blaschke products and rational functions with Siegel disks, J. Korean Math. Soc. 46 (2009), no. 1, 151-170. https://doi.org/10.4134/JKMS.2009.46.1.151
18. K. Katagata, Dynamics of rational functions with Siegel disks and polynomial semigroups, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci. 42 (2009), 17-57.
19. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, 1995.
20. K. Linda and Z. Gaofei, Bounded type Siegel disks of a one dimensional family of entire functions, preprint.
21. C. T. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math. 180 (1998), no. 2, 247-292. https://doi.org/10.1007/BF02392901
22. W. de Melo and S. van Strien, One-dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 25, Springer-Verlag, Berlin, 1993.
23. J. Milnor, Dynamics in One Complex Variable, Vieweg, 2nd edition, 2000.
24. S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics 66, 2000.
25. C. L. Petersen, On holomorphic critical quasi-circle maps, Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1739-1751. https://doi.org/10.1017/S0143385704000392
26. N. Steinmetz, Rational Iteration, Complex analytic dynamical systems, de Gruyter Studies in Mathematics, 16, Walter de Gruyter & Co., Berlin, 1993.
27. M. Yampolsky and S. Zakeri, Mating Siegel quadratic polynomials, J. Amer. Math. Soc. 14 (2001), no. 1, 25-78. https://doi.org/10.1090/S0894-0347-00-00348-9
28. S. Zakeri, Old and new on quadratic Siegel disks, http://www.math.qc.edu/-zakeri/papers/papers.
29. S. Zakeri, Dynamics of cubic Siegel polynomials, Comm. Math. Phys. 206 (1999), no. 1, 185-233. https://doi.org/10.1007/s002200050702
30. S. Zakeri, On Siegel Disks of a Class of Entire Maps, http://www.math.qc.edu/-zakeri/papers/papers.html.