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A renormalization approach to invariant circles in area-preserving maps. (English) Zbl 1194.37068
Summary: Kadanoff and Shenker introduced a renormalisation approach to invariant circles in area-preserving maps. This paper makes more precise the connection between invariant circles and the renormalisation operator. Restricting attention to noble rotation numbers, the stability of a simple fixed point of the renormalisation is analysed, corresponding to a linear twist map. It is found to be essentially attracting, so that noble circles persist under perturbation, giving a new view on KAM theory. Shenker and Kadanoff found evidence for another fixed point, corresponding to a map with a non-smooth noble circle. Further evidence is given in this paper. It has essentially only one unstable direction, and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle. Finally, noble circles are shown to be locally most robust, in an important sense.

MSC:
37E10 Dynamical systems involving maps of the circle
37E40 Dynamical aspects of twist maps
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[1] Sinclair, R.M.; Hosea, J.C.; Sheffield, G.V., Rev. sci. instruments, 41, 1552, (1970)
[2] White, R.B., (), IAEA-CN-41/T-3
[3] Arnold, V.I., Mathematical methods of classical mechanics, (1978), Springer New York · Zbl 0386.70001
[4] Birkhoff, G.D., (), 60, 111, (1932), reprinted in
[5] Nitecki, Z., Differentiable dynamics, (1971), MIT press Cambridge, Mass
[6] Moser, J.K., Stable and random motions, (1973), Princeton Univ. Press Princeton
[7] Niven, I., Irrational numbers, (), no. 11 · Zbl 0070.27101
[8] Chirikov, B.V., Phys. repts, 52, 263, (1979)
[9] Mather, J.N., Non-existence of invariant circles, (1982), preprint, Princeton
[10] Percival, I.C., Chaotic boundary of a Hamiltonian map, Physica, 6D, 67, (1982) · Zbl 1194.37061
[11] Kadanoff, L.P., Phys. rev. lett., 47, 1641, (1981)
[12] Shenker, S.J.; Kadanoff, L.P., J. stat. phys., 27, 631, (1982)
[13] Feigenbaum, M.J.; Kadanoff, L.P.; Shenker, S.J., Quasiperiodicity in dissipative systems: a renormalisation analysis, Physica, 5D, 370, (1982)
[14] Rand, D.; Ostlund, S.; Sethna, J.; Siggia, E.D., Phys. rev. lett., 49, 132, (1982), and Physica D (submitted)
[15] Escande, D.F.; Doveil, F., Phys. lett., J. stat. phys., 26, 257, (1981)
[16] Herman, M.R., Demonstration du théorème des courbes translatées de nombres de rotation de type constant, (1981), manuscript, Paris, and Les Houches notes
[17] Rüssman, H., On the existence of invariant curves of twist mappings of an annulus, (1981), preprint, Mainz, Germany
[18] Gallavotti, G., Perturbation theory of classical Hamiltonian systems, (), to appear in · Zbl 0362.46045
[19] MacKay, R.S., Renormalisation in area preserving maps, (), Princeton · Zbl 0791.58002
[20] Devaney, R., Trans. AMS, 218, 89, (1976)
[21] Krasnosel’skii, M.A.; Vainikko, G.M.; Zabreiko, P.P.; Rutitskii, Ya.B.; Stetsenko, V.Ya., Approximate solution of operator equations, (1972), Wolters-Noordhoff Groningen · Zbl 0231.41024
[22] Mather, J.N., A criterion for the non-existence of invariant circles, (1982), preprint, Princeton
[23] Greene, J.M., J. math. phys., 20, 1183, (1979)
[24] Birkhoff, G.D., 1927, dynamical systems, AMS colloq. publ., vol. 9, (1966), revised · Zbl 0171.05402
[25] Greene, J.M., J. math. phys., 9, 760, (1968)
[26] Greene, J.M., Annals of New York acad. sci., 357, 80, (1980)
[27] Schmidt, G., Phys. rev., 22A, 2849, (1980)
[28] Schmidt, G.; Bialek, J., Physica, 5D, 397, (1982)
[29] Mather, J.N., Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology, 21, 457, (1982) · Zbl 0506.58032
[30] Kadanoff, L.P., (), in press
[31] Lanford, O.E., A computer assisted proof of the Feigenbaum conjectures, (1981), IHES, preprint
[32] Eckmann, J.-P.; Koch, H.; Wittwer, P., A computer-assisted proof of universality for area preserving maps, (1982), preprint UGVA-DPT 1982/04-345
[33] Prasad, A.V., J. London math. soc., 23, 169, (1948)
[34] Greene, J.M., Nonlinear dynamics and the beam-beam interactions, (), 257
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