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Dynamics of holomorphic correspondences

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Author(s):
Carlos Alberto Siqueira Lima
Total Authors: 1
Document type: Doctoral Thesis
Press: São Carlos.
Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação
Defense date:
Examining board members:
Daniel Smania Brandão; Carlos Alberto Maquera Apaza; Sylvain Philippe Pierre Bonnot; Samuel Anton Senti
Advisor: Daniel Smania Brandão
Abstract

We generalize the notions of structural stability and hyperbolicity for the family of (multivalued) complex maps Hc(z) = zr + c; where r > 1 is rational and zr = exp r log z: We discovered that Hc is structurally stable at every hyperbolic parameter satisfying the escaping condition. Surprisingly, there may be infinitely many attracting periodic points for Hc. The set of such points gives rise to the dual Julia set, which is a Cantor set coming from a Conformal Iterated Funcion System. Both the Julia set and its dual are projections of holomorphic motions of dynamical systems (single valued maps) defined on compact subsets of Banach spaces, denoted by Xc and Wc, respectively. For c close to zero: (1) we show that Jc is a union of quasiconformal arcs around the unit circle; (2) the set Xc is an holomorphic motion of the solenoid X0; (3) using the formalism of Gibbs states we exhibit an upper bound for the Hausdorff dimension of Jc; which implies that Jc has zero Lebesgue measure. (AU)

FAPESP's process: 10/17397-2 - Dynamics of holomorphic correspondences
Grantee:Carlos Alberto Siqueira Lima
Support type: Scholarships in Brazil - Doctorate