Some transcendental entire functions with irrationally indifferent fixed points. (English) Zbl 1503.30072

Kisaka, Masashi; Naba, Hiroto

Kodai Math. J. 45, No. 3, 369-387 (2022).

Summary: Let \(S\) be the set of all transcendental entire functions of the form \(P(z) \exp (Q(z))\), where \(P\) and \(Q\) are polynomials. In this paper, by using the theory of polynomial-like mappings, we construct various kinds of functions in \(S\) with irrationally indifferent fixed points as follows:

(1)

We construct functions in \(S\) with bounded type Siegel disks centered at points other than the origin bounded by quasicircles containing critical points. This is an extension of S. Zakeri’s result in [Duke Math. J. 152, No. 3, 481–532 (2010; Zbl 1196.37085)] for \(f \in S\).

(2)

We construct functions in \(S\) with Cremer points whose multipliers satisfy some H. Cremer’s condition in [Math. Ann. 98, 151–163 (1927; JFM 53.0303.04)] only for rational functions. Our method shows that this condition can be applicable even in some transcendental cases.

(3)

For any integer \(d \geq 2\) and some \(c \in\mathbf{C} \setminus \{0\}\), we show that the function of the form \(e^{2\pi i \theta}z(1 + cz)^{d-1}e^z\,(\theta \in\mathbf{R} \backslash\mathbf{Q})\) has a Siegel point at the origin if and only if \(\theta\) is a Brjuno number. This is an extension of L. Geyer’s result in [Trans. Am. Math. Soc. 353, No. 9, 3661–3683 (2001; Zbl 0980.30017)].

(4)

For the function of the form \((e^{2\pi i\theta}z+\alpha z^2)e^z \,(\theta \in\mathbf{R} \backslash\mathbf{Q}, \,\alpha \in\mathbf{C} \backslash \{0\})\), we show that if \(\alpha\) and \(\theta\) satisfy some condition, then the Siegel disk centered at the origin is bounded by a Jordan curve containing a critical point, which is not a quasicircle. Moreover, we can choose \(\alpha\) and \(\theta\) so that the Lebesgue measure of the Julia set is positive and can also choose them so that it is zero. This is an extension of L. Keen and G. Zhang’s result in [Ergodic Theory Dyn. Syst. 29, No. 1, 137–164 (2009; Zbl 1203.37078)].

MSC:

30D20	Entire functions of one complex variable (general theory)
30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets

Keywords:

entire functions; fixed points; Cremer points; quasiconformal surgery; Siegel disks

Citations:

Zbl 1196.37085; Zbl 0980.30017; Zbl 1203.37078; JFM 53.0303.04

References:

[1]	L. V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. · Zbl 0138.06002
[2]	B. Branner and N. Fagella, Quasiconformal surgery in holomorphic dynamics, Cambridge Stud. Adv. Math. 141, Cambridge University Press, Cambridge, 2014. · Zbl 1319.37003
[3]	A. D. Brjuno, On convergence of transforms of differential equations to the normal form, Dokl. Akad. Nauk SSSR 165 (1965), 987-989. · Zbl 0146.11301
[4]	X. Buff and A. Chéritat, Quadratic Julia sets with positive area, Ann. of Math. (2) 176 (2012), 673-746. · Zbl 1321.37048
[5]	J. B. Conway, Functions of one complex variable II, Grad. Texts in Math. 159, Springer-Verlag, New York, 1995. · Zbl 0887.30003
[6]	H. Cremer, Zum Zentrumproblem, Math. Ann. 98 (1928), 151-163. · JFM 53.0303.04
[7]	H. Cremer, Über die Schrödersche Funktionalgleichung und das Schwarzsche Eckenabbildungsproblem, Math.-phys. Klasse 84 (1932), 291-324. · Zbl 0007.16902
[8]	W. Cui, Lebesgue measure of escaping sets of entire functions, Ergodic Theory Dynam. Systems 40 (2020), 89-116. · Zbl 1433.37045
[9]	A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc. Norm. Supér. (4) 18 (1985), 287-343. · Zbl 0587.30028
[10]	A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), 989-1020. · Zbl 0735.58031
[11]	L. Geyer, Siegel discs, Herman rings and the Arnold family, Trans. Amer. Math. Soc. 353 (2001), 3661-3683. · Zbl 0980.30017
[12]	J. Graczyk and G. Świątek, Siegel disks with critical points in their boundaries, Duke Math. J. 119 (2003), 189-196. · Zbl 1037.37022
[13]	L. Keen and G. Zhang, Bounded-type Siegel disks of a one-dimensional family of entire functions, Ergodic Theory Dynam. Systems 29 (2009), 137-164. · Zbl 1203.37078
[14]	J. Milnor, Dynamics in one complex variable, 3rd ed., Ann. of Math. Stud. 160, Princeton University Press, Princeton, NJ, 2006. · Zbl 1085.30002
[15]	S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic dynamics, Cambridge Stud. Adv. Math. 66, Cambridge University Press, Cambridge, 2000. · Zbl 0979.37001
[16]	Y. Okuyama, Non-linearizability of \(n\)-subhyperbolic polynomials at irrationally indifferent fixed points, J. Math. Soc. Japan 53 (2001), 847-874. · Zbl 1030.37032
[17]	C. L. Petersen, On holomorphic critical quasi-circle maps, Ergodic Theory Dynam. Systems 24 (2004), 1739-1751. · Zbl 1087.37032
[18]	C. L. Petersen and S. Zakeri, On the Julia set of a typical quadratic polynomial with a Siegel disk, Ann. of Math. (2) 159 (2004), 1-52. · Zbl 1069.37038
[19]	H. Rüssmann, Über die Iteration analytischer Funktionen, J. Math. Mech. 17 (1967), 523-532. · Zbl 0186.47704
[20]	C. L. Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942), 607-612. · Zbl 0061.14904
[21]	M. Taniguchi, Explicit representation of structurally finite entire functions, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), 68-70. · Zbl 1024.32001
[22]	J.-C. Yoccoz, Linéarisation des germes de difféomorphismes holomorphes de \(({\mathbf C},0)\), C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 55-58. · Zbl 0668.58010
[23]	S. Zakeri, Dynamics of cubic Siegel polynomials, Comm. Math. Phys. 206 (1999), 185-233. · Zbl 0936.37019
[24]	S. Zakeri, On Siegel disks of a class of entire maps, Duke Math. J. 152 (2010), 481-532. · Zbl 1196.37085