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

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.

DOI: https://doi.org/10.4213/rm758

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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

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
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• https://doi.org/10.4213/rm758
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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. A. V. Malyutin, “Classification of the group actions on the real line and circle”, St. Petersburg Math. J., 19:2 (2008), 279–296
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
3. Navas A., “Growth of groups and diffeomorphisms of the interval”, Geom. Funct. Anal., 18:3 (2008), 988–1028
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
5. Navas A., “On the Dynamics of (Left) Orderable Groups”, Annales de l Institut Fourier, 60:5 (2010), 1685–1740
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
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
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
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
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
11. L. A. Beklaryan, “Groups of line and circle homeomorphisms. Metric invariants and questions of classification”, Russian Math. Surveys, 70:2 (2015), 203–248
12. L. A. Beklaryan, “Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems”, Sb. Math., 207:8 (2016), 1079–1099
13. Shi E., Zhou L., “Topological conjugation classes of tightly transitive subgroups of $Homeo_+{(\mathbb R)}$”, Colloq. Math., 145:1 (2016), 111–120
14. Glasner E., “Short Proofs of Theorems of Malyutin and Margulis”, Proc. Amer. Math. Soc., 145:12 (2017), 5463–5467
15. Guelman N., Rivas C., “Quasi-Invariant Measures For Some Amenable Groups Acting on the Line”, Algebr. Geom. Topol., 18:2 (2018), 1067–1076
16. L. A. Beklaryan, “Groups of line and circle homeomorphisms. Criteria for almost nilpotency”, Sb. Math., 210:4 (2019), 495–507
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
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
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