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Mat. Sb., 2011, Volume 202, Number 7, Pages 43–74 (Mi msb7708)  

This article is cited in 14 scientific papers (total in 14 papers)

One-parameter semigroups of analytic functions, fixed points and the Koenigs function

V. V. Goryainov, O. S. Kudryavtseva

The Volzhsky Institute of Humanities

Abstract: Analogues of the Berkson-Porta formula for the infinitesimal generator of a one-parameter semigroup of holomorphic maps of the unit disc into itself are obtained in the case when, along with a Denjoy-Wolff point, there also exist other fixed points. With each one-parameter semigroup a so-called Koenigs function is associated, which is a solution, common for all elements of the one-parameter semigroup, of a certain functional equation (Schröder's equation in the case of an interior Denjoy-Wolff point and Abel's equation in the case of a boundary Denjoy-Wolff point). A parametric representation for classes of Koenigs functions that takes account of the Denjoy-Wolff point and other fixed points of the maps in the one-parameter semigroup is presented.
Bibliography: 19 titles.

Keywords: one-parameter semigroup, infinitesimal generator, fixed points, fractional iterates, Koenigs function.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm7708

Full text: PDF file (615 kB)
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English version:
Sbornik: Mathematics, 2011, 202:7, 971–1000

Bibliographic databases:

UDC: 517.54
MSC: 30D05
Received: 09.03.2010

Citation: V. V. Goryainov, O. S. Kudryavtseva, “One-parameter semigroups of analytic functions, fixed points and the Koenigs function”, Mat. Sb., 202:7 (2011), 43–74; Sb. Math., 202:7 (2011), 971–1000

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. Kudryavtseva O.S., “Drobnoe iterirovanie analiticheskikh v edinichnom kruge funktsii s veschestvennymi koeffitsientami”, Vestn. Volgogradskogo gos. un-ta. Ser. 1. Matem. Fiz., 2011, no. 2, 50–62  elib
    2. V. V. Goryainov, “Semigroups of analytic functions in analysis and applications”, Russian Math. Surveys, 67:6 (2012), 975–1021  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. O. S. Kudryavtseva, “Funktsiya Kenigsa i drobnoe iterirovanie analiticheskikh v edinichnom kruge funktsii s veschestvennymi koeffitsientami i nepodvizhnymi tochkami”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:1(2) (2013), 67–71  mathnet  crossref
    4. M. D. Contreras, S. Díaz-Madrigal, P. Gumenyuk, “Local duality in Loewner equations”, J. Nonlinear Convex Anal., 15:2 (2014), 269–297  mathscinet  zmath  isi  elib
    5. F. Bracci, M. D. Contreras, S. Díaz-Madrigal, P. Gumenyuk, “Boundary regular fixed points in Loewner theory”, Ann. Mat. Pura Appl. (4), 194:1 (2015), 221–245  crossref  mathscinet  zmath  isi  scopus
    6. 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
    7. 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
    8. R.-Yu. Chen, Z.-H. Zhou, “Parametric representation of infinitesimal generators on the polydisk”, Complex Anal. Oper. Theory, 10:4 (2016), 725–735  crossref  mathscinet  zmath  isi  scopus
    9. 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
    10. O. S. Kudryavtseva, “Analog of the Löwner–Kufarev Equation for the Semigroup of Conformal Mappings of the Disk into Itself with Fixed Points and Invariant Diameter”, Math. Notes, 102:2 (2017), 289–293  mathnet  crossref  crossref  mathscinet  isi  elib
    11. Gumenyuk P., “Parametric Representation of Univalent Functions With Boundary Regular Fixed Points”, Constr. Approx., 46:3 (2017), 435–458  crossref  mathscinet  zmath  isi  scopus
    12. Gumenyuk P. Prokhorov D., “Value Regions of Univalent Self Maps With Two Boundary Fixed Points”, Ann. Acad. Sci. Fenn. Ser. A1-Math., 43:1 (2018), 451–462  crossref  mathscinet  zmath  isi  scopus
    13. D. V. Prokhorov, “Value regions in classes of conformal mappings”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 19:3 (2019), 258–279  mathnet  crossref  elib
    14. O. S. Kudryavtseva, A. P. Solodov, “Two-sided estimates for domains of univalence for classes of holomorphic self-maps of a disc with two fixed points”, Sb. Math., 210:7 (2019), 1019–1042  mathnet  crossref  crossref  adsnasa  isi  elib
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