New Discretization of Complex Analysis: The Euclidean and Hyperbolic Planes

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 (see [1]). 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. Introduction. History. We do not discuss here ”geometric” discretizations of conformal mappings started in early XX century. Our goal is to discretize Cauchy-Riemann operator ∂̄ as a linear difference operator. It was done first time by Lelong-Ferrand in 1940 (see [2]). Her discretization is based on the square lattice in R. Discrete version L of ∂̄ acts on the C-valued functions ψ of vertices Lψ(m,n) = ψ(m,n) + iψ(m+ 1, n)− ψ(m+ 1, n+ 1)− iψ(m,n + 1) The equation Lψ = 0 defines d-holomorphic functions. Many people developed this approach (see recent literature in [4]). Let me point out on the two weak points in it: First, it is in fact a second order difference operator because two length scales are involved in the sum (lengths of the side and diagonal). Second, there is no natural factorization similar to ∆ = ∂̄∂, in the square lattice. Discrete holomorphic functions do not form algebra neither in the classical approach nor in our new approach based on simplicial complexes. Sergey P. Novikov, IPST and MATH Department, University of Maryland, College Park MD, USA and Landau Institute, Moscow, e-mail [email protected]; This work is partially supported by the Russian Grant in the Nonlinear Dynamics.