Abstract
Let I be the group of rotations of the circle and for each n, let I n be the subgroup of I having exactly n elements. For sufficiently small ε, it is shown that every ε-homomorphism from I n into I is an ε-perturbation of a homomorphism. Best possible results are given for n=2, 3, 4. For maps from I n into I n , best possible results are given for n≤12.
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References
CenzerD., ‘The Stability Problem for Transformations of the Circle’, Proc. Royal Soc. Edinburgh 84A, 279–281 (1979).
De laHarpeP. and KaroubiM., ‘Representations approchees d'un groupe dans une algebre de Banach’, Manuscripta Math. 22, 293–310 (1977).
HyersD. H., ‘On the Stability of the Linear Functional Equation’, Proc. Nat. Acad. Sci. U.S.A. 27, 222–224 (1941).
ShapiroH. N., ‘Note on a Problem in Number Theory’, Bull. Amer. Math. Soc. 54, 890–893 (1948).
Ulam, S., Sets, Numbers and Universes, MIT Press, 1974.
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Cenzer, D. The stability problem: New results and counterexamples. Lett Math Phys 10, 155–160 (1985). https://doi.org/10.1007/BF00398152
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DOI: https://doi.org/10.1007/BF00398152