Separatrix diagram

Claim 2.1.4 (Fomenko, Zieschang [289], [298]).. Let Q e H), that iSf is a constant-energy surface of an integrable system (by a Bott integral f). Let m be the number of stable periodic solutions of the system, s the number of unstable periodic solutions with nonorientable separatrix diagram, and r the number of critical Klein bottles. Then we always have the following inequalities ... 

Since, for the moment, the separatrix diagram of the critical circle ia supposed to be orientable, the first case (the Mobius strip) is impossible here. The tubular neighbourhood of the surface P. is therefore homeomorphic to a round handle. It is glued to Ca-g precisely in the fashion suggested by the definition of the round-handle -gluing operation (see above). The axes of both feet of the round handle are glued to two smooth circles 71 and 72 drawn on Ba-g by points A and B (Fig. 27) when the point x slides upon S. ... 

Separatrix Diagrams Cut Out Nontrivial Cycles on Nonsingular LiouvUle Tori... 

The tubular neighbourhood of the separatrix diagram P. in the nonorientable case will be called a thickened (or thick) Mobius strip. 

When passing through the critical value a, the torus splits into two tori, Ti g and T2, . The separatrix diagram P. goes from the critical surface and, when descending, meets the torus along two circles 71 and 72 (see above). Lemmas... 

Lemma 2.1.9. l) Let be a critical saddle circle and let its separatrix diagram P be orientable. Then a three-dimensional manifold C(S ) with the boundary Ti g U T2, U T-g is homeomorphic to a direct product x 5, where is... 

Now consider the nonorientahle case. The boundary circle of a separatrix diagram P.( Mobius strip) is glued to a torus 2L along some of its generators... 

Tone Handles. A Separatrix Diagram Is Always Glued to a Nonsingular Liou-ville Torus T Along a Nontrivial (n — 1)-Dimensional Cycle T ... 

If a critical torus has the dimension n, then it is either the set of the local minimum or of the local maximum of the energy H. In this case, either two close nonsingular Liouville tori flow into one torus or the torus T splits into two tori r. Let P2 = P2 T ) and = P T ) be, respectively, in and out separatrix diagrams of the critical submanifold... 

Setting the element a determines a certain number k of generators in the critical torus going round which a normal segment of the separatrix diagram P ... 

The reasoning will be more delicate. The point is that in this case the parallelepiped n itself is insufficient. Indeed, the definition of a nonorientable separatrix diagram implies that the orbits... 

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