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

This article is cited in 15 scientific papers (total in 15 papers)

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)
References: PDF file   HTML file

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
Received: 24.09.2012

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

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    Citing articles on Google Scholar: Russian citations, English citations
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    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    3. D. Betsakos, “On the rate of convergence of parabolic semigroups of holomorphic functions”, Anal. Math. Phys., 5:2 (2015), 207–216  crossref  mathscinet  zmath  isi  elib  scopus
    4. V. V. Goryainov, “Evolution families of conformal mappings with fixed points and the Löwner-Kufarev equation”, Sb. Math., 206:1 (2015), 33–60  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. D. Betsakos, “On the rate of convergence of hyperbolic semigroups of holomorphic functions”, Bull. Lond. Math. Soc., 47:3 (2015), 493–500  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref
    7. V. N. Dubinin, “Schwarzian Derivative and Covering Arcs of a Pencil of Circles by Holomorphic Functions”, Math. Notes, 98:6 (2015), 920–925  mathnet  crossref  crossref  mathscinet  isi  elib
    8. D. Betsakos, “On the Asymptotic Behavior of the Trajectories of Semigroups of Holomorphic Functions”, J. Geom. Anal., 26:1 (2016), 557–569  crossref  mathscinet  zmath  scopus
    9. D. Betsakos, “Geometric description of the classification of holomorphic semigroups”, Proc. Amer. Math. Soc., 144:4 (2016), 1595–1604  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  scopus
    11. S. R. Nasyrov, “Uniformization of one-parametric families of complex tori”, Russian Math. (Iz. VUZ), 61:8 (2017), 36–45  mathnet  crossref  isi
    12. V. N. Dubinin, “Geometric estimates for the Schwarzian derivative”, Russian Math. Surveys, 72:3 (2017), 479–511  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. P. Gumenyuk, “Parametric representation of univalent functions with boundary regular fixed points”, Constr. Approx., 46:3 (2017), 435–458  crossref  mathscinet  zmath  isi
    14. D. Betsakos, “On the eigenvalues of the infinitesimal generator of a semigroup of composition operators”, J. Funct. Anal., 273:7 (2017), 2249–2274  crossref  mathscinet  zmath  isi
    15. Goryainov V.V., “Loewner-Kufarev Equation For a Strip With An Analogue of Hydrodynamic Normalization”, Lobachevskii J. Math., 39:6 (2018), 759–766  crossref  mathscinet  isi  scopus
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