Аннотация:
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.
Образец цитирования:
S. P. Novikov, “New discretization of complex analysis: The Euclidean and hyperbolic planes”, Современные проблемы математики, Сборник статей. К 75-летию Института, Труды МИАН, 273, МАИК «Наука/Интерпериодика», М., 2011, 257–270; Proc. Steklov Inst. Math., 273 (2011), 238–251
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\paper New discretization of complex analysis: The Euclidean and hyperbolic planes
\inbook Современные проблемы математики
\bookinfo Сборник статей. К~75-летию Института
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\vol 273
\pages 257--270
\publ МАИК «Наука/Интерпериодика»
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\jour Proc. Steklov Inst. Math.
\yr 2011
\vol 273
\pages 238--251
\crossref{https://doi.org/10.1134/S0081543811040122}
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Bobenko A.I., Bucking U., “Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals For Ramified Coverings of the Riemann Sphere”, Math. Phys. Anal. Geom., 24:3 (2021), 23
Legatiuk A., Guerlebeck K., Hommel A., “The Discrete Borel-Pompeiu Formula on a Rectangular Lattice”, Adv. Appl. Clifford Algebr., 28:3 (2018), UNSP 69
И. А. Дынников, “Ограниченные дискретные голоморфные функции на плоскости Лобачевского”, Топология и физика, Сборник статей. К 80-летию со дня рождения академика Сергея Петровича Новикова, Труды МИАН, 302, МАИК «Наука/Интерпериодика», М., 2018, 202–213; I. A. Dynnikov, “Bounded discrete holomorphic functions on the hyperbolic plane”, Proc. Steklov Inst. Math., 302 (2018), 186–197
Ulrike Bücking, Symmetries and Integrability of Difference Equations, 2017, 153
И. А. Дынников, “О новой дискретизации комплексного анализа”, УМН, 70:6(426) (2015), 63–84; I. A. Dynnikov, “On a new discretization of complex analysis”, Russian Math. Surveys, 70:6 (2015), 1031–1050
Цитирование в формате AMSBIB
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