| Budget Amount *help |
¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1998: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1997: ¥500,000 (Direct Cost: ¥500,000)
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| Research Abstract |
There is a famous problem, namely, the perturbation problem of the twist map defined on a two-dimensional annulus, which is treated in "Lectures on Celestial Mechanics" written by Siegel-Moser in 1971. Our investigation is concerned with the perturbation of a twist map which is defined not on the annulus but on the product space of the unit disk D and the Lie group G of the fractional linear transformations acting on D. We recall that the twist map t on an annulus is defined by t(r, s)=(r, r + s), where (r, s) stands for the polar coordinate of the annulus. Now we replace the annulus with the product space G x D and introduce the twist map T on G x D by T(a, z)=(a, a(z)). Since this map preserves each {a} x D for all a in G and since their union covers the whole space G x D, the dynamical system determined by T is an integrable system on G x D. Our result is as follows : if a is an elliptic element of G, and the rotation angle of a is a, Diophantine number, then the invariant set {a} x D is persistent under small perturbations of T. Here, recall that every eliptic a in G is similar to an rotation z→exp(ik)z, and we call this real number k the rotation angle of a.
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