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## Dynamical Classification of some Birational Maps of C2

This dissertation addresses three different problems in the study of discrete dynamical systems.

Firstly, this work dynamically classifies a 9−parametric family of planar birational

maps f : C2 → C2 that is

f(x, y) =

α0 + α1x + α2y,

β0 + β1x + β2y

γ0 + γ1x + γ2y

,

where the parameters are complex numbers. This is done by finding the dynamical degree δ

for the degenerate and non degenerate cases of F that is the extended map of f in projective

space. The dynamical degree δ defined as

δ(F) := lim

n→∞

(deg(Fn))

1

n ,

indicates the subfamilies which are chaotic, that is when δ > 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}.

Identifer | oai:union.ndltd.org:TDX_UAB/oai:www.tdx.cat:10803/316026 |

Date | 14 July 2014 |

Creators | Zafar, Sundus |

Contributors | Cima, Anna, Universitat Autònoma de Barcelona. Departament de Matemàtiques |

Publisher | Universitat Autònoma de Barcelona |

Source Sets | Universitat Autònoma de Barcelona |

Language | English |

Detected Language | English |

Type | info:eu-repo/semantics/doctoralThesis, info:eu-repo/semantics/publishedVersion |

Format | 192 p., application/pdf |

Source | TDX (Tesis Doctorals en Xarxa) |

Rights | L'accés als continguts d'aquesta tesi queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: http://creativecommons.org/licenses/by-nc-nd/3.0/es/, info:eu-repo/semantics/openAccess |

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