Preprint A51/2008
A Franks' lemma that preserves invariant manifolds
DDIC
Keywords: Franks' lemma | linear cocycle | invariant manifolds | strong stable | unstable | homoclinic intersection
Un c�l�bre lemme de John Franks dit que toute perturbation de la diff�rentielle d'un diff�omorphisme $f$ le long d'une orbite p�riodique peut �tre r�alis�e par une $C^1$-perturbation $g$ du diff�omorphisme sur un voisinage arbitrairement petit de ladite orbite. Ce lemme cependant ne donne aucune information sur le comportement des vari�t�s invariantes de l'orbite p�riodique apr�s perturbation. Dans cet article nous montrons que si la perturbation de la d�riv�e peut �tre jointe � la d�riv�e initiale par un chemin, alors la distance $C^1$ entre $f$ et $g$ peut �tre trouv�e arbitrairement proche du diam�tre du chemin. De plus, si des directions stables ou instables d'indices fix�s existent le long du chemin, alors les vari�t�s invariantes correspondantes peuvent �tre pr�serv�es en-dehors d'un voisinage arbitrairement petit de l'orbite. A well-known lemma by John Franks asserts that one can realise any perturbation of the derivative of a diffeomorphism $f$ along a periodic orbit by a $C^1$-perturbation $g$ of the whole diffeomorphism on an arbitrarily small neighbourhood of the periodic point. However, that lemma does not provide any information on the behaviour of the invariant manifolds of the periodic point for $g$. In this paper we show that if the perturbated derivative can be joined from the initial derivative by a continuous path, then the $C^1$-distance between $f$ and $g$ can be found arbitrarily close to the diameter of the path. Moreover, if strong stable or unstable directions of some indices exist along that path, then the corresponding invariant manifolds can be preserved outside a small neighbourhood of the orbit.