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 Uspekhi Mat. Nauk, 2012, Volume 67, Issue 6(408), Pages 5–52 (Mi umn9500)

Semigroups of analytic functions in analysis and applications

V. V. Goryainov

Volzhsky Institute of Humanities (branch) of the Volgograd State University

Abstract: This survey considers problems of analysis and certain related areas in which semigroups of analytic functions with respect to the operation of composition appear naturally. The main attention is devoted to holomorphic maps of a disk (or a half-plane) into itself. The role of fixed points is highlighted, both in the description of the structure of semigroups and in applications. Interconnections of the problem of fractional iteration with certain problems in the theory of random branching processes are pointed out, as well as with certain questions of non-commutative probability. The role of the infinitesimal description of semigroups of conformal maps in the development of the parametric method in the theory of univalent functions is shown.
Bibliography: 94 titles.

Keywords: one-parameter semigroup, infinitesimal generator, evolution family, evolution equation, fractional iterates, Koenigs function, fixed points.

DOI: https://doi.org/10.4213/rm9500

Full text: PDF file (847 kB)
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English version:
Russian Mathematical Surveys, 2012, 67:6, 975–1021

Bibliographic databases:

Document Type: Article
UDC: 517.54
MSC: Primary 30-02; Secondary 30-03, 30C20, 30C35, 30C45, 30C50, 30C75, 30D05, 39B12, 39B32, 60E05, 60E10, 60J80

Citation: V. V. Goryainov, “Semigroups of analytic functions in analysis and applications”, Uspekhi Mat. Nauk, 67:6(408) (2012), 5–52; Russian Math. Surveys, 67:6 (2012), 975–1021

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. P. Gumenyuk, “Angular and unrestricted limits of one-parameter semigroups in the unit disk”, J. Math. Anal. Appl., 417:1 (2014), 200–224
2. M. D. Contreras, S. Díaz-Madrigal, P. Gumenyuk, “Slope problem for trajectories of holomorphic semigroups in the unit disk”, Comput. Methods Funct. Theory, 15:1 (2015), 117–124
3. D. Betsakos, “On the rate of convergence of parabolic semigroups of holomorphic functions”, Anal. Math. Phys., 5:2 (2015), 207–216
4. V. V. Goryainov, “Evolution families of conformal mappings with fixed points and the Löwner-Kufarev equation”, Sb. Math., 206:1 (2015), 33–60
5. D. Betsakos, “On the rate of convergence of hyperbolic semigroups of holomorphic functions”, Bull. Lond. Math. Soc., 47:3 (2015), 493–500
6. O. S. Kudryavtseva, “Holomorphic maps of the disk into itself with invariant diameter and bounded distortion”, Russian Math. (Iz. VUZ), 59:8 (2015), 41–51
7. V. N. Dubinin, “Schwarzian Derivative and Covering Arcs of a Pencil of Circles by Holomorphic Functions”, Math. Notes, 98:6 (2015), 920–925
8. D. Betsakos, “On the Asymptotic Behavior of the Trajectories of Semigroups of Holomorphic Functions”, J. Geom. Anal., 26:1 (2016), 557–569
9. D. Betsakos, “Geometric description of the classification of holomorphic semigroups”, Proc. Amer. Math. Soc., 144:4 (2016), 1595–1604
10. F. Bracci, P. Gumenyuk, “Contact points and fractional singularities for semigroups of holomorphic self-maps of the unit disc”, J. Anal. Math., 130 (2016), 185–217
11. S. R. Nasyrov, “Uniformization of one-parametric families of complex tori”, Russian Math. (Iz. VUZ), 61:8 (2017), 36–45
12. V. N. Dubinin, “Geometric estimates for the Schwarzian derivative”, Russian Math. Surveys, 72:3 (2017), 479–511
13. P. Gumenyuk, “Parametric representation of univalent functions with boundary regular fixed points”, Constr. Approx., 46:3 (2017), 435–458
14. D. Betsakos, “On the eigenvalues of the infinitesimal generator of a semigroup of composition operators”, J. Funct. Anal., 273:7 (2017), 2249–2274
15. Goryainov V.V., “Loewner-Kufarev Equation For a Strip With An Analogue of Hydrodynamic Normalization”, Lobachevskii J. Math., 39:6 (2018), 759–766
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