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On a family of rational perturbations of the doubling map
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The goal of this paper is to investigate the parameter plane of a rational family of perturbations of the doubling map given by the Blaschke products $B_a(z)=z^3\frac{z-a}{1-\bar{a}z}$. First we study the basic properties of these maps such as the connectivity of the Julia set as a function of the parameter $a$. We use techniques of quasiconformal surgery to explore the relation between certain members of the family and the degree 4 polynomials $\left(\overline{\overline{z}^2+c}\right)^2+c$. In parameter space, we classify the different hyperbolic components according to the critical orbits and we show how to parametrize those of disjoint type.
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CANELA SÁNCHEZ, Jordi, FAGELLA RABIONET, Núria and GARIJO REAL, Antonio. On a family of rational perturbations of the doubling map. Journal of Difference Equations and Applications. 2015. Vol. 21, num. 8, pags. 715-741. ISSN 1023-6198. [consulted: 16 of August of 2026]. Available at: https://hdl.handle.net/2445/67324