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 Mat. Sb., 1996, Volume 187, Number 4, Pages 3–28 (Mi msb121)

On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. II. Projectively-invariant measures

L. A. Beklaryan

Central Economics and Mathematics Institute, RAS

Abstract: Groups of orientation-preserving homeomorphisms of $\mathbb R$ are studied. Such metric invariants as projectively-invariant measures are investigated. The approach taken results in the classification of groups of homeomorphisms by the complexity of the set of all fixed points of the group elements. In each of the classes of groups thus distinguished a finer classification is carried out in terms of the complexity of the topological structure of orbits and the combinatorial properties of the group.

DOI: https://doi.org/10.4213/sm121

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English version:
Sbornik: Mathematics, 1996, 187:4, 469–494

Bibliographic databases:

UDC: 515.168.3
MSC: Primary 54H15, 58F11; Secondary 28D05, 20F38

Citation: L. A. Beklaryan, “On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. II. Projectively-invariant measures”, Mat. Sb., 187:4 (1996), 3–28; Sb. Math., 187:4 (1996), 469–494

Citation in format AMSBIB
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• https://doi.org/10.4213/sm121
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This publication is cited in the following articles:
1. L. A. Beklaryan, “A criterion connected with the structure of the fixed-point set for the existence of a projectively invariant measure for groups of orientation-preserving homeomorphisms of $\mathbb R$”, Russian Math. Surveys, 51:3 (1996), 539–540
2. P. de la Harpe, R. I. Grigorchuk, T. Ceccherini-Silberstein, “Amenability and Paradoxical Decompositions for Pseudogroups and for Discrete Metric Spaces”, Proc. Steklov Inst. Math., 224 (1999), 57–97
3. Beklaryan, LA, “omega-projectively invariant measures for the groups of orientation-preserving homeomorpfisms of line”, Doklady Akademii Nauk, 367:6 (1999), 727
4. L. A. Beklaryan, “On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. III. $\omega$-projectively invariant measures”, Sb. Math., 190:4 (1999), 521–538
5. L. A. Beklaryan, “On a criterion for the topological conjugacy of a quasisymmetric group to a group of affine transformations of $\mathbb R$”, Sb. Math., 191:6 (2000), 809–819
6. L. A. Beklaryan, “On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line”, Math. Notes, 71:3 (2002), 305–315
7. L. A. Beklaryan, “Introduction to the theory of functional differential equations and their applications. Group approach”, Journal of Mathematical Sciences, 135:2 (2006), 2813–2954
8. 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
9. Bleak C., Kassabov M., Matucci F., “Structure Theorems for Groups of Homeomorphisms of the Circle”, Internat J Algebra Comput, 21:6 (2011), 1007–1036
10. L. A. Beklaryan, “Criteria for the Existence of an Invariant Measure for Groups of Homeomorphisms of the Line”, Math. Notes, 95:3 (2014), 304–307
11. 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
12. L. A. Beklaryan, “Groups of line and circle homeomorphisms. Metric invariants and questions of classification”, Russian Math. Surveys, 70:2 (2015), 203–248
13. L. A. Beklaryan, “Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems”, Sb. Math., 207:8 (2016), 1079–1099
14. L. A. Beklaryan, “Groups of line and circle homeomorphisms. Criteria for almost nilpotency”, Sb. Math., 210:4 (2019), 495–507
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