Constructing an expanding metric for dynamical systems in one complex variable

We describe a rigorous computer algorithm for attempting to construct an explicit, discretized metric for which a polynomial map is expansive on a neighbourhood of the Julia set, J. We show that the construction of such a metric proves the map is hyperbolic. We also examine the question of whether the algorithm can be improved, and the related question of how to build a metric close to Euclidean. Finally, we give several examples generated with our implementation of this algorithm.

[1]  Suzanne Lynch Hruska.  Rigorous numerical models for the dynamics of complex Henon mappings on their chain recurrent sets , 2004, math/0406002.

[2]  Suzanne Lynch Hruska.  A numerical method for proving hyperbolicity of complex Henon mappings , 2004 .

[3]  G. Osipenko.  Symbolic Image, Hyperbolicity, and Structural Stability , 2003 .

[4]  Konstantin Mischaikow,et al.  Topological techniques for efficient rigorous computation in dynamics , 2002, Acta Numerica.

[5]  G. Osipenko.  Spectrum of a dynamical system and applied symbolic Dynamics , 2000 .

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

[7]  George Osipenko,et al.  Applied symbolic dynamics: attractors and filtrations , 1998 .

[8]  J. Milnor.  Dynamics in One Complex Variable: Introductory Lectures , 1990, math/9201272.

[9]  F. Przytycki.  Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map , 1985 .

[10]  Anthony Manning,et al.  The dimension of the maximal measure for a polynomial map , 1984 .

[11]  H. Brolin.  Invariant sets under iteration of rational functions , 1965 .

[12]  Michael Dellnitz,et al.  Chapter 5 - Set Oriented Numerical Methods for Dynamical Systems , 2002 .

[13]  Warwick Tucker,et al.  Foundations of Computational Mathematics a Rigorous Ode Solver and Smale's 14th Problem , 2022 .

[14]  George Osipenko,et al.  Construction of attractors and filtrations , 1999 .

[15]  Konstantin Mischaikow,et al.  Exploring global dynamics: a numerical algorithm based on the conley index theory , 1995 .

[16]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[17]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.