Volume 8 (2008), Number 2. Abstracts

M. Braverman and M. Yampolsky. Computability of Julia Sets [PDF]

In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a polynomial is always computable.

Keywords. Julia set, computability, complexity.

2000 Mathematics Subject Classification. Primary: 37F50; Secondary: 68Q17.

Yu. V. Brezhnev. On Uniformization of Algebraic Curves [PDF]

Based on Burnside's parametrization of the algebraic curve y² = x⁵−x we obtain the remaining attributes of its uniformization: associated Fuchsian equations and their solutions, accessory parameters, monodromies, conformal maps, fundamental polygons, etc. As a generalization, we propose a way of uniformization of arbitrary curves by zero genus groups. In the hyperelliptic case all the objects of the theory are explicitly described. We consider a large number of examples and, briefly, applications: Abelian integrals, metrics of Poincaré, differential equations of the Jacobi–Chazy and Picard–Fuchs type, and others.

Keywords. Uniformization of algebraic curves, Riemann surfaces, Fuchsian equations/groups, monodromy groups, accessory parameters, modular equations, conformal maps, curvelinear polygons, θ-functions, Abelian integrals, metrics of Poincaré, moduli spaces.

2000 Mathematics Subject Classification. Primary 30F10; 30F35.

I. Fesenko. Adelic Approach to the Zeta Function of Arithmetic Schemes in Dimension Two [PDF]

This paper suggests a new approach to the study of the fundamental properties of the zeta function of a model of elliptic curve over a global field. This complex valued commutative approach is a two-dimensional extension of the classical adelic analysis of Tate and Iwasawa. We explain how using structures which come naturally from the explicit two-dimensional class field theory and working with a new R((X))-valued translation invariant measure, integration theory and harmonic analysis on various complete objects associated to arithmetic surfaces one can define and study zeta integrals which are closely related to the zeta function of a regular model of elliptic curve over global fields. In the two-\hskip0pt dimensional adelic analysis the study of poles of the zeta function is reduced to the study of poles of a boundary term which is an integral of a certain arithmetic function over the boundary of an adelic space. The structure of the boundary and function determines the analytic properties of the boundary term and location of the poles of the zeta function, which results in applications of the theory to several key directions of arithmetic of elliptic curves over global fields.

Keywords. Elliptic curves over global fields, arithmetic surfaces, zeta function, zeta integral, two-dimensional adelic spaces, harmonic analysis, Hasse zeta functions, analytic duality, boundary term, meromorphic continuation and functional equation, mean-periodic functions, Laplace–Carleman transform, generalized Riemann hypothesis, Birch and Swinnerton–Dyer conjecture, automorphic representations.

2000 Mathematics Subject Classification. 14G10, 11M99, 19F05, 11G40, 11G99, 11F99, 43-99, 11M45, 44A10, 46N99.

J. Ribón. Modulus of Analytic Classification for Unfoldings of Resonant Diffeomorphisms [PDF]

We provide a complete system of analytic invariants for unfoldings of non-linearizable resonant complex analytic diffeomorphisms as well as its geometrical interpretation. In order to fulfill this goal we develop an extension of the Fatou coordinates with controlled asymptotic behavior in the neighborhood of the fixed points. The classical constructions are based on finding regions where the dynamics of the unfolding is topologically stable. We introduce a concept of infinitesimal stability leading to Fatou coordinates reflecting more faithfully the analytic nature of the unfolding. These improvements allow us to control the domain of definition of a conjugating mapping and its power series expansion.

Keywords. Resonant diffeomorphism, analytic classification, bifurcation theory, structural stability.

2000 Mathematics Subject Classification. Primary: 37F45; Secondary: 37G10, 37F75.

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