By Claude Godbillon (auth.)
These notes are an elaboration of the 1st a part of a direction on foliations which i've got given at Strasbourg in 1976 and at Tunis in 1977. they're involved commonly with dynamical sys tems in dimensions one and , particularly so that it will their purposes to foliated manifolds. a tremendous bankruptcy, although, is lacking, which might were facing structural balance. The ebook of the French variation was once re alized by-the efforts of the secretariat and the printing workplace of the dept of arithmetic of Strasbourg. i'm deeply thankful to all those that contributed, specifically to Mme. Lambert for typing the manuscript, and to Messrs. Bodo and Christ for its copy. Strasbourg, January 1979. desk of Contents I. VECTOR FIELDS ON MANIFOLDS 1. Integration of vector fields. 1 2. basic conception of orbits. thirteen three. Irlvariant and minimaI units. 18 four. restrict units. 21 five. course fields. 27 A. Vector fields and isotopies. 34 II. THE neighborhood BEHAVIOUR OF VECTOR FIELDS 39 1. balance and conjugation. 39 2. Linear differential equations. forty four three. Linear differential equations with consistent coefficients. forty seven four. Linear differential equations with periodic coefficients. 50 five. version box of a vector box. fifty two 6. Behaviour close to a novel aspect. fifty seven 7. Behaviour close to a periodic orbit. fifty nine A. Conjugation of contractions in R. sixty seven III. PLANAR VECTOR FIELDS seventy five 1. restrict units within the aircraft. seventy five 2. Periodic orbits. eighty two three. Singular issues. ninety four. The Poincare index.
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PROPOSITION. The produet of the eharaeteristie numbers of a periodie orbit y of X equals exp Jydiv X dt. (Let M be an orientable manifold and w a volume form on M. ) 56 Proof. The relation MX R determines a smooth function ft(X) on for which we find af t at W=ddt(ep;w) = ep: (Ixw) = (divX°Cllt)ftw. Thus f If, however, T exp (x) t Jot . d~v X (ep (x» dt. 9. The case of surfaces. D. denote a periodic orbit of a vector field X without singularity on an orientable surface M. ) For avolume form W on M the Pfaffian form 0/ = i w has no x singularities, and it is possible to find another Pfaffian form (i: tisfying 0/ W= 0/ A ei thus have of y dO/ = is exp ( - ä(x) = -1.
Denoting by (~ t ) the flow generated by X we obtain a homeomor- phism h of the open unit ball D of center 0 onto the ball B by setting h(tx)=~ -Log t (sx)forxt:S m-l andtt:(O,l), h(O) = 0, and this homeomorphism is a conjugation from YID to XIB. D. 5. Remarks. 2l. It is, however, a diffeomorphism exeept at the origin. 4 shows that the field X is topologieally eonjugate to its variational equation near y, provided all eharaeteristie values of X in y have strietly negative (respeetively positive) real parts.
Under these conditions we have: i) the set of all solutions of (F) on J is an affine space having S as its underlying vector space. ii) For a fundamental matrix solution X(t) of (E) we look for a solution of (F) of the form y(t) = X(t)c(t). rhe solution of (F) with 47 initial eondition y(t o ) = yo is thus found to be y (t) X(t)X(t) -1 o y 0 + X(t) fX(S) - lb (s)dS. 'his proeedure is known as "variation of the eonstants". 3. 1. Sueh equations are investigated by transforming A into its Jordan form [10].