It is well known that the integrability (solvability) of a differential equation is related to the singularity structure of its solutions in the complex domain - an observation that lies behind the Painleve test. A number of ways of extending this philosophy to discrete equations are explored. First, following the classical work of Julia, Birkhoff and others, a natural interpretation of these equations in the complex domain as difference or delay equations is described and it is noted that arbitrary periodic functions play an analogous role for difference equations to that played by arbitrary constants in the solution of differential equations. These periodic functions can produce spurious branching in solutions and are factored out of the analysis which concentrates on branching from other sources. Second, examples and theorems from the theory of difference equations are presented which show that, modulo these periodic functions, solutions of a large class of difference equations are meromorphic, regardless of their integrability. It is argued that the integrability of many difference equations is related to the structure of their solutions at infinity in the complex plane and that Nevanlinna theory provides many of the concepts necessary to detect integrability in a large class of equations. A perturbative method is then constructed and used to develop series in z and the derivative of log (z ), where z is the independent variable of the difference equation. This method provides an analogue of the series developed in the Painleve test for differential equations. Finally, the implications of these observations are discussed for two tests which have been studied in the literature regarding the integrability of discrete equations.

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On the extension of the Painlevé property to difference equations

This is a tutorial on some of the main ideas in KAM theory. The goal is to present the background and to explain and compare somewhat informally some of the main methods of proof. It is an expanded version of the lectures given by the author in the Summer Research Institute on Smooth Ergodic Theory Seattle, 1999. The style is pedagogical and expository and it only aims to be an introduction to the primary literature. It does not aim to be a systematic survey nor to present full proofs.

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A Tutorial on Kam Theory

Let f(z) = ez+ z, where θ is an irrational number of bounded type. According to Siegel, f is linearizable on a disk containing the origin. In this paper we show: • the Hausdorff dimension of the Julia set J(f) is strictly less than two; and • if θ is a quadratic irrational (such as the golden mean), then the Siegel disk for f is self-similar about the critical point. In the latter case, we also show the rescaled first-return maps converge exponentially fast to a system of commuting branched coverings of the complex plane.

/pdf/self-similarity-of-siegel-disks-and-hausdorff-dimension-of-4c4ptwuk42.pdf

Self-similarity of Siegel disks and Hausdorff dimension of Julia sets

The Painlev´ e property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlev´ e equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlev´ e property for discrete equations, in which the independent variable is taken to be complex. A general introduction to Nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q-difference operators. New results are presented concerning equations of the form w(z +1 )w(z − 1) = R(z, w), where R is rational in w with meromorphic coefficients.

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Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

The Fatou set FR for a rational map R: C--*C is the set of points z c C possessing a neighbourhood on which the family of iterates {R n }n~>o is normal (in the sense of Montel). The Julia set JR=C--FR is the complement of the Fatou set. (The monographs [CG], [Be], [St] provide introductions to the theory of iteration of rational maps.) Let 0E ]0, 1 [ Q be an irrational number and write it as a continued fraction

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麻美鈴木

Papers

「On the Iteration of Analytic Functions」(木村俊房先生の仕事から)

麻美 鈴木

Papers

「On the Iteration of Analytic Functions」(木村俊房先生の仕事から)

麻美鈴木