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
ω-projectively invariant measures), and the main
‘obstructions’ to the existence of metric invariants of this kind
are described.
Citation:
L. A. Beklaryan, “Groups of homeomorphisms of the line and the circle.
Topological characteristics and metric invariants”, Russian Math. Surveys, 59:4 (2004), 599–660
\Bibitem{Bek04}
\by L.~A.~Beklaryan
\paper Groups of homeomorphisms of the line and the circle.
Topological characteristics and metric invariants
\jour Russian Math. Surveys
\yr 2004
\vol 59
\issue 4
\pages 599--660
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\crossref{https://doi.org/10.1070/RM2004v059n04ABEH000758}
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Linking options:
https://www.mathnet.ru/eng/rm758
https://doi.org/10.1070/RM2004v059n04ABEH000758
https://www.mathnet.ru/eng/rm/v59/i4/p3
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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
L. A. Beklaryan, “On massive subsets in the space of finitely generated groups of diffeomorphisms of the line and the circle in the case of C(1) smoothness”, J. Math. Sci., 257:6 (2021), 780–796
Guelman N., Rivas C., “Quasi-Invariant Measures For Some Amenable Groups Acting on the Line”, Algebr. Geom. Topol., 18:2 (2018), 1067–1076
Glasner E., “Short Proofs of Theorems of Malyutin and Margulis”, Proc. Amer. Math. Soc., 145:12 (2017), 5463–5467
L. A. Beklaryan, “Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems”, Sb. Math., 207:8 (2016), 1079–1099
Shi E., Zhou L., “Topological conjugation classes of tightly transitive subgroups of Homeo+(R)”, Colloq. Math., 145:1 (2016), 111–120
L. A. Beklaryan, “Groups of line and circle homeomorphisms. Metric invariants and questions of classification”, Russian Math. Surveys, 70:2 (2015), 203–248
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
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
L. A. Beklaryan, “Residual Subsets in the Space of Finitely Generated Groups of Diffeomorphisms of the Circle”, Math. Notes, 93:1 (2013), 29–35
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
Bleak C., Kassabov M., Matucci F., “Structure Theorems for Groups of Homeomorphisms of the Circle”, Internat J Algebra Comput, 21:6 (2011), 1007–1036
Navas A., “On the Dynamics of (Left) Orderable Groups”, Annales de l Institut Fourier, 60:5 (2010), 1685–1740
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
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
Navas A., “Growth of groups and diffeomorphisms of the interval”, Geom. Funct. Anal., 18:3 (2008), 988–1028
A. V. Malyutin, “Classification of the group actions on the real line and circle”, St. Petersburg Math. J., 19:2 (2008), 279–296
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