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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 4(358), Pages 3–68 (Mi umn758)  

This article is cited in 18 scientific papers (total in 18 papers)

Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants

L. A. Beklaryan

Central Economics and Mathematics Institute, RAS

Abstract: This survey is devoted to investigations concerning topological, algebraic, and combinatorial characteristics as well as metric invariants for arbitrary groups of homeomorphisms of the line and the circle. Relationships between these characteristics are established, the most important metric invariants are studied (in the form of invariant, projectively invariant, and $\omega$-projectively invariant measures), and the main ‘obstructions’ to the existence of metric invariants of this kind are described.


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English version:
Russian Mathematical Surveys, 2004, 59:4, 599–660

Bibliographic databases:

UDC: 512.544.43
MSC: Primary 54H15; Secondary 28D05, 28C10, 37C40, 37E10, 37A99
Received: 08.04.2004

Citation: L. A. Beklaryan, “Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants”, Uspekhi Mat. Nauk, 59:4(358) (2004), 3–68; Russian Math. Surveys, 59:4 (2004), 599–660

Citation in format AMSBIB
\by L.~A.~Beklaryan
\paper Groups of homeomorphisms of the line and the circle.
Topological characteristics and metric invariants
\jour Uspekhi Mat. Nauk
\yr 2004
\vol 59
\issue 4(358)
\pages 3--68
\jour Russian Math. Surveys
\yr 2004
\vol 59
\issue 4
\pages 599--660

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    This publication is cited in the following articles:
    1. A. V. Malyutin, “Classification of the group actions on the real line and circle”, St. Petersburg Math. J., 19:2 (2008), 279–296  mathnet  crossref  mathscinet  zmath  isi  elib
    2. Bastos M.A., Fernandes C.A., Karlovich Yu.I., “$C^*$-algebras of singular integral operators with shifts having the same nonempty set of fixed points”, Complex Anal. Oper. Theory, 2:2 (2008), 241–272  crossref  mathscinet  zmath  isi  elib
    3. Navas A., “Growth of groups and diffeomorphisms of the interval”, Geom. Funct. Anal., 18:3 (2008), 988–1028  crossref  mathscinet  zmath  isi
    4. Wang S., Shi E., Zhou L., Cairns G., “Topological Transitivity of Solvable Group Actions on the Line R”, Colloquium Mathematicum, 116:2 (2009), 203–215  crossref  mathscinet  zmath  isi
    5. Navas A., “On the Dynamics of (Left) Orderable Groups”, Annales de l Institut Fourier, 60:5 (2010), 1685–1740  crossref  mathscinet  zmath  isi
    6. Bleak C., Kassabov M., Matucci F., “Structure Theorems for Groups of Homeomorphisms of the Circle”, Internat J Algebra Comput, 21:6 (2011), 1007–1036  crossref  mathscinet  zmath  isi
    7. L. A. Beklaryan, “Residual Subsets in the Space of Finitely Generated Groups of Diffeomorphisms of the Circle”, Math. Notes, 93:1 (2013), 29–35  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. Beklaryan L.A., “Group Specialties in the Problem of the Maximum Principle for Systems with Deviating Argument”, J. Dyn. Control Syst., 18:3 (2012), 419–432  crossref  mathscinet  zmath  isi  elib
    9. Aziz Belmiloudi, “Dynamical Behavior of Nonlinear Impulsive Abstract Partial Differential Equations on Networks with Multiple Time-Varying Delays and Mixed Boundary Conditions Involving Time-Varying Delays”, J Dyn Control Syst, 2014  crossref  mathscinet  isi
    10. L. A. Beklaryan, “Groups of homeomorphisms of the line. Criteria for the existence of invariant and projectively invariant measures in terms of the commutator subgroup”, Sb. Math., 205:12 (2014), 1741–1760  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. L. A. Beklaryan, “Groups of line and circle homeomorphisms. Metric invariants and questions of classification”, Russian Math. Surveys, 70:2 (2015), 203–248  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    12. L. A. Beklaryan, “Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems”, Sb. Math., 207:8 (2016), 1079–1099  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. Shi E., Zhou L., “Topological conjugation classes of tightly transitive subgroups of $Homeo_+{(\mathbb R)}$”, Colloq. Math., 145:1 (2016), 111–120  crossref  mathscinet  zmath  isi  scopus
    14. Glasner E., “Short Proofs of Theorems of Malyutin and Margulis”, Proc. Amer. Math. Soc., 145:12 (2017), 5463–5467  crossref  mathscinet  zmath  isi
    15. Guelman N., Rivas C., “Quasi-Invariant Measures For Some Amenable Groups Acting on the Line”, Algebr. Geom. Topol., 18:2 (2018), 1067–1076  crossref  mathscinet  zmath  isi
    16. L. A. Beklaryan, “Groups of line and circle homeomorphisms. Criteria for almost nilpotency”, Sb. Math., 210:4 (2019), 495–507  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    17. L. A. Beklaryan, “O massivnykh podmnozhestvakh v prostranstve konechno porozhdënnykh grupp diffeomorfizmov pryamoi i okruzhnosti v sluchae gladkosti $C^{(1)}$”, Fundament. i prikl. matem., 22:4 (2019), 51–74  mathnet
    18. Dirbak M., Hric R., Malicky P., Snoha L'ubomir, Spitalsky V., “Minimality For Actions of Abelian Semigroups on Compact Spaces With a Free Interval”, Ergod. Theory Dyn. Syst., 39:11 (2019), PII S0143385718000044, 2968–2982  crossref  isi
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