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An entire transcendental family with a persistent Siegel disc. (English) Zbl 1198.30026

The goal in this paper is to describe the dynamics of the one parameter family of entire transcendental maps
\[ f_a(z) = \lambda a\big(e^{z/a}(z + 1 - a) - 1 + a\big), \]
where \(a\in\mathbb C\setminus\{0\}\) and \(\lambda = e^{2\pi i\theta}\) with \(\theta\) being a fixed irrational Brjuno number. Observe that 0 is a fixed point of the multiplier \(\lambda\) and therefore, for all values of the parameter \(a\), there is a persistent Siegel disc \(\Delta_a\) around \(z=0\). The functions \(f_a\) have two singular values: the image of the only critical point \(w =-1\) and an asymptotic value at \(v_a = \lambda a(a- 1)\), which has one and only one finite pre-image at the point \(p_a = a- 1\). The authors investigate the stable components of the parameter plane (capture components and semi-hyperbolic components) and also some topological properties of the Siegel disc in terms of the parameter.

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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