New Discretization of Complex Analysis: The Euclidean and Hyperbolic Planes

TL;DR

This paper extends a discretization approach of complex analysis from Euclidean to hyperbolic planes, revealing new dynamical phenomena and solving fundamental boundary problems using symbolic dynamics.

Contribution

It introduces a new discretization of complex analysis for hyperbolic planes, building on previous Euclidean lattice work, and explores novel dynamical phenomena.

Findings

Development of a DCA theory for hyperbolic plane lattices

Discovery of interesting dynamical phenomena in hyperbolic DCA

Application of symbolic dynamics to boundary problems

Abstract

Few years ago we developed jointly with I.Dynnikov new discretization of complex analysis (DCA) based on the two-dimensional manifolds with colored black/white triangulation. Especially deep results were obtained for the Euclidean plane with equilateral triangle lattice. In the present work we develop a DCA theory for the analogs of equilateral triangle lattice in Hyperbolic plane. Some specific very interesting "dynamical phenomena" appear in this case solving most fundamental boundary problems. Mike Boyle from the University of Maryland helped to use here the methods of symbolic dynamics