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An entire transcendental family with a persistent Siegel disc

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We study the class of entire transcendental maps of finite order with one critical point and one asymptotic value, which has exactly one finite pre-image, and having a persistent Siegel disc. After normalisation this is a one parameter family $f_{a}$ with $a \in \mathbb{C}^{*}$ which includes the semi-standard map $\lambda z \mathrm{e}^{z}$ at $a=1$, approaches the exponential map when $a \rightarrow 0$ and a quadratic polynomial when $a \rightarrow \infty$. We 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.

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Journal of Difference Equations and Applications, 2010, vol. 16, num. 5, p. 523-553

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Articles publicats en revistes (Matemàtiques i Informàtica)

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An entire transcendental family with a persistent Siegel disc