Geodesic
New discretization of complex analysis: The Euclidean and hyperbolic planes
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, Tome 273 (2011), pp. 257-270.

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Discretization of complex analysis on the plane based on the standard square lattice was started in the 1940s. It was developed by many people and also extended to the surfaces subdivided by the squares. In our opinion, this standard discretization does not preserve well-known remarkable features of the completely integrable system. These features certainly characterize the standard Cauchy continuous complex analysis. They played a key role in the great success of complex analysis in mathematics and applications. Few years ago, jointly with I. Dynnikov, we developed a new discretization of complex analysis (DCA) based on the two-dimensional manifolds with colored black/white triangulation. Especially profound results were obtained for the Euclidean plane with an equilateral triangle lattice. Our approach preserves a lot of features of completely integrable systems. In the present work we develop a DCA theory for the analogs of an equilateral triangle lattice in the hyperbolic plane. This case is much more difficult than the Euclidean one. Many problems (easily solved for the Euclidean plane) have not been solved here yet. Some specific very interesting “dynamical phenomena” appear in this case; for example, description of boundaries of the most fundamental geometric objects (like the round ball) leads to dynamical problems. Mike Boyle from the University of Maryland helped me to use here the methods of symbolic dynamics. 
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S. P. Novikov. New discretization of complex analysis: The Euclidean and hyperbolic planes. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, Tome 273 (2011), pp. 257-270. http://geodesic.mathdoc.fr/item/TM_2011_273_a11/

[1] Dynnikov I.A., Novikov S.P., “Geometry of the triangle equation on two-manifolds”, Moscow Math. J., 3:2 (2003), 419–438 | MR | Zbl

[2] Ferrand J., “Fonctions preharmoniques et fonctions preholomorphes”, Bull. sci. math. Ser. 2, 68 (1944), 152–180 | MR | Zbl

[3] Duffin R.J., “Basic properties of discrete analytic function”, Duke Math. J., 23 (1956), 335–363 | DOI | MR | Zbl

[4] Novikov S.P., “Diskretnye svyaznosti i raznostnye lineinye uravneniya”, Tr. MIAN, 247, 2004, 186–201 | MR | Zbl

[5] Bobenko A.I., Mercat C., Suris Yu.B., “Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function”, J. reine und angew. Math., 583 (2005), 117–161 | DOI | MR | Zbl

[6] Dolbilin N.P., Shtanko M.A., Shtogrin M.I., “Kubicheskie podkompleksy v pravilnykh reshetkakh”, DAN SSSR, 291:2 (1986), 277–279 | MR

[7] Dolbilin N.P., Sedrakyan A.G., Shtanko M.A., Shtogrin M.I., “Topologiya semeistva parametrizatsii dvumernykh tsiklov, voznikayuschikh v trekhmernoi modeli Izinga”, DAN SSSR., 295:1 (1987), 19–23 | MR | Zbl

[8] Dolbilin N.P., Shtanko M.A., Shtogrin M.I., “Problema parametrizatsii tsiklov po modulyu 2 v trekhmernoi kubicheskoi reshetke”, Izv. AN SSSR. Ser. mat., 52:2 (1988), 355–377 | MR | Zbl

[9] Dolbilin N.P., Shtanko M.A., Shtogrin M.I., “Kvadrilyazhi i parametrizatsii reshetchatykh tsiklov”, Tr. MIAN, 196, 1991, 66–85 | MR

[10] Dolbilin N.P., Shtanko M.A., Shtogrin M.I., “Kubicheskie mnogoobraziya v reshetkakh”, Izv. RAN. Ser. mat., 58:2 (1994), 93–107 | MR | Zbl

[11] Novikov S.P., Topology, Encycl. Math. Sci., 12, Topology I, Springer, Berlin, 1996 | MR | Zbl

[12] Novikov S.P., “Algebraicheskie svoistva dvumernykh raznostnykh operatorov”, UMN, 52:1 (1997), 225–226 | DOI | MR | Zbl

[13] Novikov S.P., Dynnikov I.A., “Diskretnye spektralnye simmetrii malomernykh differentsialnykh operatorov i raznostnykh operatorov na pravilnykh reshetkakh i dvumernykh mnogoobraziyakh”, UMN, 52:5 (1997), 175–234 | DOI | MR | Zbl

[14] Grinevich P.G., Novikov R.G., “Yadro Koshi dlya DN-diskretnogo kompleksnogo analiza Novikova–Dynnikova na treugolnoi reshetke”, UMN, 62:4 (2007), 155–156 | DOI | MR | Zbl