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Dynamical Classification of some Birational Maps of C2
Zafar, Sundus
Cima, Anna, dir.
Universitat Autònoma de Barcelona. Departament de Matemàtiques

Publicació: [Barcelona] : Universitat Autònoma de Barcelona, 2015
Descripció: 192 p.
Nota: Tesi doctoral - Universitat Autònoma de Barcelona. Departament de Matemàtiques, 2014 1, and otherwise. The study of the sequence of degrees dn of F shows the degree growth rate of all the subfamilies of f. This gives the families which have bounded growth , or they grow linearly, quadratically or grow exponentially. The family f includes the birational maps studied by Bedford and Kim in [18] as one of its subfamily. The second problem includes the study of the subfamilies of f with zero entropy that is for δ = 1. These includes the families with bounded (in particular periodic), linear or quadratic growth rate. Two transverse fibrations are found for the families with bounded growth. In the periodic case the period of the families is indicated. It is observed that there exist infinite periodic subfamilies of f, depending on the parameter region. The families with linear growth rate preserve rational fibration and the quadratic growth rate families preserve elliptic fibration that is unique depending on the parameters. In all the cases with zero entropy all the mappings are found up to affine conjugacy. Thirdly, it deals with non-autonomous Lyness type recurrences of the form xn+2 = an + xn+1 xn , where {an}n is a k-periodic sequence of complex numbers with minimal period k. We treat such non-autonomous recurrences via the autonomous dynamical system generated by the birational mapping Fak ◦ Fak−1 ◦ · · · ◦ Fa1 where Fa is defined by Fa(x, y) = (y, a + y x ). For the cases k ∈ {1, 2, 3, 6} the corresponding mappings have a rational first integral. By calculating the dynamical degree we show that for k = 4 and for k = 5 generically the dynamical system is no longer rationally integrable. We also prove that the only values of k for which the corresponding dynamical system is rationally integrable for all the values of the involved parameters, are k ∈ {1, 2, 3, 6}.
Drets: L'accés als continguts d'aquesta tesi queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons Creative Commons
Llengua: Anglès.
Document: Tesis i dissertacions electròniques ; doctoralThesis ; publishedVersion
Matèria: Birational maps of C2 ; Picard Group ; Discrete dynamical systems
ISBN: 9788449048029

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192 p, 1.8 MB

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