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Extra info for Celestial mechanics. Vol. 5, Part 2. Topology of the three-body problem.

Example text

1 is a translation arc, since its extremities correspond to one another by S and this segment does not contain any other pair of points corresponding to each other by S. The ,{ are the only translation arcs on the segment l. The initial part of the line b = BB 1B 2 • • • constructed by the above process is contained in the rectangle between 10 and /1 • The line b cannot abut on the line y = 1 or y = 2, because of the assumption. An essential feature of the construction is that any finite part of the plane contains only a finite number of the base-points of the construction.

Thus the a-theorem is established. The hypotheses of the theorem stated at the beginning include those of the a-Theorem, and in addition we may exclude the alternative (b) of the a-Theorem for any positive a. Hence for every positive a there exists a point P of R which is carried by T into a point T(P) of R 1 on the same radial line and distant from P by not more than a. A sequence of such points P with a approaching zero evidently has at least one limiting point in R and R 1, which is invariant under T.

That a circle through s- 1(0), 0, S(O) cuts E is the condition for S to be hyperbolic. A circle C with its center at s- 1( oo) is orthogonal to E. The part of the x-sphere, that is, the region inside and outside the unit circle, constitutes the fundamental domain for an infinite number of cyclic subgroups of r formed by powers of S, which we denote by {S}. If we join each point of C to the corresponding points on S(C) by S, then a torus is obtained. If the same point on the torus is made to correspond to the points obtained by the operations of {S} from a point of the fundamental domain, then the x-sphere described at the base-point is the covering surface of the torus.

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