Abstract
Finitely generated groups of diffeomorphisms of the circle are considered in different problems of geometry, wave theory, variational calculus, etc. In particular, the set of such groups contains groups freely acting on the orbits of almost each point of the circle. The present paper deals with the structure of the set of finitely generated groups of diffeomorphisms with a given number of generators and with the above property. It is shown that such a set contains a residual subset (i.e., contains the countable intersection of open everywhere dense subsets).
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L. A. Beklaryan, “A variational problem with retarded argument, and its connection with a certain semigroup of mappings of an interval into itself,” Dokl. Akad. Nauk SSSR 271(5), 1036–1040 (1983) [Soviet Math. Dokl. 28, 186–190 (1983)].
L. A. Beklaryan, “An optimal control problem for systems with deviating argument, and its connection with the finitely generated group of homeomorphisms of R generated by the deviation functions,” Dokl. Akad. Nauk SSSR 317(6), 1289–1294 (1991) [Soviet Math. Dokl. 43 (2), 600–605 (1991)].
L. A. Beklaryan, “About canonical types of the differential equations with deviating argument,” Funct. Differ. Equ., No. 1–2, 25–33 (2001).
L. A. Beklaryan, “Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants,” UspekhiMat. Nauk 59(4), 3–68 (2004) [RussianMath. Surveys 59 (4), 599–660 (2004)].
L. A. Beklaryan, An Introduction to the Theory of Functional Differential Equations: Group Approach (Faktorial Press, Moscow, 2007) [in Russian].
V. I. Arnol’d, Additional Chapters in the Theory of Ordinary Differential Equations (Nauka, Moscow, 1978) [in Russian].
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Original Russian Text © L. A. Beklaryan, 2013, published in Matematicheskie Zametki, 2013, Vol. 92, No. 6, pp. 825–833.
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Beklaryan, L.A. Residual subsets in the space of finitely generated groups of diffeomorphisms of the circle. Math Notes 93, 29–35 (2013). https://doi.org/10.1134/S0001434613010033
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DOI: https://doi.org/10.1134/S0001434613010033