A Perturbation Lemmas In this appendix we review certain general perturbation statements that are at the basis of most genericity results mentioned in the text. So far these perturbation lemmas have been proved in the C1 -topology only. That is the reason why, at this stage, a nice description of generic diffeomorphisms is available only in that topology. Pugh's closing lemma (Section A.I) has a very intuitive statement which gives no hint of the complexity of its proof: any orbit going back arbitrarily close to itself can be closed by a C1 -small perturbation of the map. The proof contains two main ingredients. The first one is to estimate the effect of small perturbations of the diffeomorphism on a small neighborhood of a finite segment of orbit. The second one is to select two returns of the orbit close to itself which are, in some sense, closer to one another than to any intermediary returns. See also Section A.4. Each of this arguments has been generalized, leading important new per- turbations lemmas. For the ergodic closing lemma (Section A.2), Mane con- sidered how frequently, one may take the initial point as one of the selected returns. The reason why this question is important is that the periodic orbit one creates by closing an orbit segment in this way has derivative close to that of the original orbit segment. Hayashi's connecting lemma (Section A.3) was originally designed as a tool for creating intersections between invariant manifolds of periodic points. For that purpose, the strategy was to shorten in a systematic way (Pugh's perturbations) orbit segments almost joining the two invariant manifolds. See Section A.4 for further comments on the proof In Section A.6 we state a very useful simple result of Franks, allowing to realize perturbations of the derivative along of periodic obits by C1 pertur- bations of the actual dynamical systems. This argument turns certain per- turbations problems into problems about how eigenspaces and eigenvalues of products of matrices depend on small perturbations of the factors. 278 A Perturbation Lemmas A.I Closing lemmas A typical question is: Given a recurrent point of a system, is there a Cr nearby system for which that point is periodic? The difficulty of the question depends, dramatically, on the topology and, to some extent, on the dimension of the ambient space and the class of systems one considers. For r = 0 the answer is positive and easy. For r = 1 it is still positive, as stated by Pugh's closing lemma: Theorem A.I (Pugh [363, 364]). Let f = {/* : t G E} be a C1 flow on a compact manifold M, and x G M be a non-wandering point for this flow: there exist Xj —-> x and times tj —-> -f-oo such that ftj[xj) —> x. Then there exist flows g = {gf : t G M} arbitrarily close to f in the C1 topology, for which x is a periodic point. Partial results had been obtained by Peixoto [351], for flows on orient able surfaces, and Anosov [16], for the class of systems named after him. Actually, in the special case of hyperbolic systems this problem has a complete answer, known as the shadowing lemma: a closing orbit exists in the system itself, no perturbation is needed. Theorem A.I was extended by Pugh, Robinson [365] to several other classes of systems, including C1 diffeomorphisms, Hamiltonian flows, symplectic dif- feomorphisms, and volume preserving diffeomorphisms and flows. See also [23] for recent proofs based on a different approach. Let us mention that the C1 closing lemma remains open in the setting of geodesic flows (C2 perturbation of the Riemannian metric). For r > 1 the question of the closing lemma remains very much open. Gutierrez [203] showed that local perturbations as used by Pugh in the C1 case do not suffice if r > 2. Most important, Herman [214, 215] constructed counterexamples to the C°° closing lemma in the setting of conservative sys- tems, symplectic diffeomorphisms or Hamiltonian vector fields, in sufficiently high dimensions. More precisely, give any r > 2n > 4, in [214] he exhibits Cr open sets of Hamiltonian functions on T2n, such that the corresponding Hamiltonian vector fields l on the 2n-torus, endowed with a constant symplec- tic form, exhibit open sets formed by energy surfaces that contain no periodic orbit. In fact, on each component of these energy surfaces the dynamics is C1 conjugate to a linear irrational (Diophantine) flow. See also [454]. On the other hand, no counterexamples are known in the most interesting mechanical case, that is, when the ambient space is the cotangent bundle T*M of some symplectic manifold (M,u;), endowed with the natural symplectic form. In view of these negative results, it is convenient to look for weaker useful versions of the closing lemma. The result of Peixoto mentioned above holds for any 0 < r < oo. It suggests that a perhaps more realistic goal could be 2The Hamiltonian vector field X corresponding to a function H on a symplectic manifold (A4» is defined by H(X, •) = ViJ. If H is Cr then X is C"1. A.3 Connecting lemmas 279 Problem A.2. Given a Cr system with a recurrent point x, is there an ar- bitrarily Cr close system for which either x is periodic or else there are no recurrent points in a neighborhood of xl On the other hand, the closing lemma has been refined in two important directions, always in the C1 topology: A.2 Ergodic closing lemma In many situations one would like to know: Does the periodic orbit obtained after perturbation remain close, up to the period, to a segment of the original orbit? Mane's ergodic closing lemma asserts that this is true for a full proba- bility set of points, that is, having full measure with respect to any invariant probability of the unperturbed map. For the precise statement, let S(f) be the set of points x G M such that given any C1 neighborhood U of f and any e > 0 there exist g £lA, coinciding with / outside the ^-neighborhood of the /-orbit of x, and there exists a ^-periodic point y £ M such that d(fi(x),gi(y)) <e for all 0 < i < per(y). Theorem A.3 (Mane [278]). For any C1 diffeomorphism f of a compact manifold, the set S(f) has full probability for f. A major application of this statement is to express in terms of periodic orbits dynamical behaviors that are detected by invariant measures. The fol- lowing argument, which is part of Mane's proof of the 2-dimensional version of Theorem 7.5 in [278], is a good illustration: Let / be a surface diffeomorphism with a dominated splitting E 0 F and suppose the sub-bundle F is not uniformly expanding. Then there are arbi- trarily long orbit segments over which expansion along F is arbitrarily weak. Any weak accumulation point of Dirac averages along such orbit segments is an invariant measure for which the sub-bundle F is not expanding. Pugh's closing lemma permits to close a typical orbit of such a measure. However, not knowing whether the periodic orbit remains close to the original one, one loses control on the dilation along F. On the contrary, the ergodic closing lemma ensures that the derivative of the perturbed map g is, at best, weakly expanding along the continuation Fg of F for g over the orbit of the periodic point y. By another small perturbation (this is Franks' lemma, that we shall discuss in a while) we can now turn any slight expansion into a contraction, which means that y becomes a sink. A.3 Connecting lemmas The connecting lemma is motivated by the following question. Assume that the unstable manifold of a periodic point accumulates on the stable manifold 280 A Perturbation Lemmas of another (possibly the same) periodic point. Is there a small perturbation of the dynamical system for which the two invariant manifolds corresponding to the continuations of the two periodic points actually intersect each other? While this is closely related to the problem of the closing lemma, there is one subtle difference. In the context of the closing lemma it suffices to create a periodic orbit passing close to the initial recurrent point x: once that is done, a change of coordinates close to the identity allows us to suppose that the periodic point is x itself. In other words, points near x are indistinguishable from each other. This is clearly no longer true in the setting of the connecting lemma, because now one wants to create orbits connecting invariant (stable and unstable) manifolds that are given a priori. Even in the simplest case, flows on orient able surfaces, this problem re- mained open for more than three decades after the corresponding closing problem had been settled. This was also a major difficulty in the proof of the stability conjecture for diffeomorphisms [283], where a main step consists in showing that if the map is not hyperbolic then homoclinic connections may be created by small C1 perturbations. In [282], Mane bypassed the lack, at the time, of a connecting lemma by proving several connecting results with addi- tional assumptions: in a few words, he supposed that orbits accumulating the two invariant manifolds spent a positive fraction of time near the associated periodic point. y q P Fig. A.I. Connecting invariant manifolds Eventually, the problem was solved by Hayashi [208] for C1 diffeomor- phisms and flows. In fact he proves a stronger statement, cf.