Computable Caratheodory Theory

Abstract The conformal Riemann mapping of the unit disk onto a simply-connected domain W is a central object of study in classical Complex Analysis. The first complete proof of the Riemann Mapping Theorem given by P. Koebe in 1912 is constructive, and theoretical aspects of computing the Riemann map have been extensively studied since. Caratheodory Theory describes the boundary extension of the Riemann map. In this paper we develop its constructive version with explicit complexity bounds.

[1]  S. Kakutani.  143. Two-dimensional Brownian Motion and Harmonic Functions , 1944 .

[2]  Mark Braverman,et al.  Non-computable Julia sets , 2004, ArXiv.

[3]  Thomas Ransford,et al.  Potential Theory in the Complex Plane: Bibliography , 1995 .

[4]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[5]  S. E. Warschawski.  On the degree of variation in conformal mapping of variable regions , 1950 .

[6]  Mark Braverman,et al.  Derandomization of Euclidean Random Walks , 2007, APPROX-RANDOM.

[7]  Timothy H. McNicholl.  An Effective Carathéodory Theorem , 2011, Theory of Computing Systems.

[8]  C. Pommerenke.  Uniformly perfect sets and the Poincaré metric , 1979 .

[9]  Donald E. Marshall,et al.  Harmonic Measure: Jordan Domains , 2005 .

[10]  Mark Braverman,et al.  Computability of Julia Sets , 2009, Algorithms and computation in mathematics.

[11]  Peter Hertling,et al.  The Effective Riemann Mapping Theorem , 1999, Theor. Comput. Sci..

[12]  J. Milnor,et al.  Dynamics in One Complex Variable: Introductory Lectures , 2000 .

[13]  Klaus Weihrauch,et al.  Computable Analysis: An Introduction , 2014, Texts in Theoretical Computer Science. An EATCS Series.

[14]  P. Koebe.  Methoden der konformen abbildung und uniformisierung , 1929 .

[15]  Two . dimensional Brownian Motion and Harmonic Functions , 2022 .

[16]  Mark Braverman,et al.  On the computational complexity of the Riemann mapping , 2005, math/0505617.

[17]  J. Milnor.  Dynamics in one complex variable , 2000 .