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 Trudy Mat. Inst. Steklova, 2011, Volume 273, Pages 257–270 (Mi tm3277)

New discretization of complex analysis: The Euclidean and hyperbolic planes

S. P. Novikovabc

a Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
b Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, Russia
c Institute for Physical Science and Technology, Department of Mathematics, University of Maryland, College Park, MD, USA

Abstract: 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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English version:
Proceedings of the Steklov Institute of Mathematics, 2011, 273, 238–251

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UDC: 517.548.8
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Citation: S. P. Novikov, “New discretization of complex analysis: The Euclidean and hyperbolic planes”, Modern problems of mathematics, Collected papers. In honor of the 75th anniversary of the Institute, Trudy Mat. Inst. Steklova, 273, MAIK Nauka/Interperiodica, Moscow, 2011, 257–270; Proc. Steklov Inst. Math., 273 (2011), 238–251

Citation in format AMSBIB
\Bibitem{Nov11} \by S.~P.~Novikov \paper New discretization of complex analysis: The Euclidean and hyperbolic planes \inbook Modern problems of mathematics \bookinfo Collected papers. In honor of the 75th anniversary of the Institute \serial Trudy Mat. Inst. Steklova \yr 2011 \vol 273 \pages 257--270 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3277} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2893550} \zmath{https://zbmath.org/?q=an:05963537} \elib{https://elibrary.ru/item.asp?id=16456350} \transl \jour Proc. Steklov Inst. Math. \yr 2011 \vol 273 \pages 238--251 \crossref{https://doi.org/10.1134/S0081543811040122} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000295982500012} 

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This publication is cited in the following articles:
1. I. A. Dynnikov, “On a new discretization of complex analysis”, Russian Math. Surveys, 70:6 (2015), 1031–1050
2. Legatiuk A., Guerlebeck K., Hommel A., “The Discrete Borel-Pompeiu Formula on a Rectangular Lattice”, Adv. Appl. Clifford Algebr., 28:3 (2018), UNSP 69
3. I. A. Dynnikov, “Bounded discrete holomorphic functions on the hyperbolic plane”, Proc. Steklov Inst. Math., 302 (2018), 186–197
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