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Singular perturbations in the quadratic family with multiple poles
Garijo, Antoni (Universitat Rovira i Virgili. Departament d’Enginyeria Informàtica i Matemàtiques)
Marotta, Sebastian M. (University of the Pacific(Stockon). Department of Mathematics)
Russell, Elizabeth D. (United States Military Academy West Point. Department of Mathematical Sciences)

Fecha: 2013
Resumen: We consider the quadratic family of complex maps given by qc(z) = z2 + c where c is the center of a hyperbolic component in the Mandelbrot set. Then, we introduce a singular perturbation on the corresponding bounded superattracting cycle by adding one pole to each point in the cycle. When c = −1 the Julia set of q−1 is the well known basilica and the perturbed map is given by fλ(z) = z2 − 1 + λ/(z d0 (z + 1)d1) where d0, d1 ≥ 1 are integers, and λ is a complex parameter such that $� � $ is very small. We focus on the topological characteristics of the Julia and Fatou sets of fλ that arise when the parameter λ becomes nonzero. We give sufficient conditions on the order of the poles so that for small λ the Julia sets consist of the union of homeomorphic copies of the unperturbed Julia set, countably many Cantor sets of concentric closed curves, and Cantor sets of point components that accumulate on them.
Nota: Número d'acord de subvenció AGAUR/2009/SGR-792
Nota: Número d'acord de subvenció MINECO/MTM2008-01486
Nota: Agraïments: The first author is partially supported by the European Community through the project 035651-1-2-CODY.
Derechos: Tots els drets reservats.
Lengua: Anglès
Documento: article ; recerca ; preprint
Materia: Complex dynamical systems ; Dynamics of rational maps
Publicado en: Journal of Difference Equations and Applications, Vol. 19 (2013) , p. 124-145, ISSN 1023-6198

DOI: 10.1080/10236198.2011.630668

25 p, 932.3 KB

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