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On Siegel disks of a class of entire maps. (English) Zbl 1196.37085

Let \(P\) and \(Q\) be polynomials of degree \(p\geq 0\) and \(q\geq 0\), respectively, and suppose that \(f:z\mapsto P(z)\exp(Q(z))\) has a Siegel disk at \(0\) with rotation number \(\theta\). The main result states that if \(\theta\) is of bounded type, then the boundary of such a Siegel disk is a quasi-circle that contains a critical point of \(f\). Since the degrees of the polynomials \(P\) and \(Q\) are allowed to be arbitrary, this theorem generalizes and unifies in an elegant way various results on boundaries of Siegel disks for polynomials and certain transcendental entire maps, such as \(z\mapsto \text{e}^{2\pi i \theta} z\text{e}^z\) and \(z\mapsto (\text{e}^{2\pi i \theta} z + a z^2) \text{e}^z\).
The restriction to Siegel disks at \(0\) is crucial; in fact, the statement of the main theorem is not true if the center of the Siegel disk is allowed to be arbitrary, as examples in the family \(z\mapsto \lambda \text{e}^{z-\lambda}\) show [compare M.-R. Herman, Commun. Math. Phys. 99, 593–612 (1985; Zbl 0587.30040)]. In the transcendental case, \(f\) has a unique asymptotic value at \(0\) and the normalization \(f(0)=0\) prescribes its dynamics, and each iterate of \(f\) has again the same unique asymptotic value. This restriction is crucial in order to approach the functions with quasi-conformal surgery, which is a standard method for this kind of problems and is also the main tool this work is based on.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)

Citations:

Zbl 0587.30040
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References:

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