On computational complexity of Siegel Julia sets

It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high.Comment: Updated version, to appear in Commun. Math. Phy

Technical Report: Computation on the Extended Complex Plane and Conformal Mapping of Multiply-connected Domains

AbstractWe introduce a system for computation on the extended complex plane based on the Type-Two Effectivity approach to computable analysis. Included are computations on meromorphic functions, open sets, and closed sets. Applications to Möbius transformations, boundaries of multiply connected domains, and conformal mapping of multiply connected domains are considered

The connection between computability of a nonlinear problem and its linearization: the Hartman-Grobman theorem revisited

As one of the seven open problems in the addendum to their 1989 book Computability in Analysis and Physics Pour-El and Richards (1989)[17], Pour-El and Richards asked, "What is the connection between the computability of the original nonlinear operator and the linear operator which results from it?" Yet at present, systematic studies of the issues raised by this question seem to be missing from the literature. In this paper, we study one problem in this direction: the Hartman-Grobman linearization theorem for ordinary differential equations (ODEs). We prove, roughly speaking, that near a hyperbolic equilibrium point x(0) of a nonlinear ODE (x) over dot = f(x), there is a computable homeomorphism H such that H circle phi = L circle H, where phi is the solution to the ODE and L is the solution to its linearization (x) over dot = Df (x(0)) x. (C) 2012 Elsevier B.V. All rights reserved.Fundacao para a Ciencia e a Tecnologia; EU FEDER POCTI/POCI via SQIG - Instituto de Telecomunicacoes through the FCT [PEst-OE/EEI/LA0008/2011