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Hausdorff dimension of a limit set for a family of nonholomorphic perturbations of the map \(z \to z^2\). (English) Zbl 0934.37045

The author considers a family of maps \(f_{\lambda,C}: \mathbb{C}\to \mathbb{C}\) which are defined by one of the following formulas: \[ f_{\lambda,C}= \begin{cases} |z|^{2\lambda-2}\cdot z^2+C\\ z^{\lambda+1}\cdot\overline{z}^{\lambda-1}+C\\ r^{2\lambda} e^{2i\varphi},\quad z=re^{i\varphi} \end{cases} \] for parameters \(\frac 12< \lambda<1\) and \(|C|\neq 0\). Note that \(f_{\lambda,C}\) is a nonholomorphic map of \(\mathbb{C}\) and can be considered as perturbations of \(z\mapsto z^2\). The author proves that the Hausdorff dimension of the limit set for iterations of \(f_{\lambda,C}\) (in all three cases, see above) is greater than one.

MSC:

37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
28A80 Fractals
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