Generic Properties of Dynamical Systems Christian Bonatti

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Christian Bonatti. Generic Properties of Dynamical Systems. Encyclopedia of Mathematical Physics, Academic Press, 2006, pp. 494-502. 10.1016/B0-12-512666-2/00164-4. hal-01348445

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Distributed under a Creative Commons Attribution| 4.0 International License Generic Properties of Dynamical Systems represented as a point x in a geometrical space Rn. C Bonatti, Université de Bourgogne, Dijon, France In many cases, the quantities describing the state are related, so that the phase space (space of all possible states) is a submanifold M Rn. The time evolution R Introduction of the system is represented by a curve xt, t 2 drawn on the phase space M, or by a sequence xn 2 The state of a concrete system (from physics, M, n 2 Z, if we consider discrete time (i.e., every chemistry, ecology, or other sciences) is described day at the same time, or every January 1st). using (finitely many, say n) observable quantities Believing in determinism, and if the system is (e.g., positions and velocities for mechanical isolated from external influences, the state x0 of the systems, population densities for echological system at the present time determines its evolution. systems, etc.). Hence, the state of a system may be For continuous-time systems, the infinitesimal

1 evolution is given by a differential equation or vector generic properties, generic diffeomorphisms verify field dx=dt = X(x); the vector X(x) represents velo- simultanuously all the properties Pi. city and direction of the evolution. For a discrete-time A property P is C r-robust if the set of diffeo- system, the evolution rule is a function F : M ! M;if morphisms verifying P is open in Diff r(M). A x is the state at time t,thenF(x) is the state at the property P is locally generic if there is an (nonempty) time t þ 1. The evolution of the system, starting at open set O on which it is generic, that is, there is the initial data x0, is described by the orbit of x0,that residual set R such that P is verified on R\O. is, the sequence {(xn)n2Z j xnþ1 = F(xn)} (discrete The properties of generic dynamical systems time) or the maximal solution xt of the differential depend mostly on the dimension of the manifold M equation ax=dt = X(x) (continuous time). and of the C r-topology considered, r 2 N [ {þ1} (an important problem is that Cr-generic diffeo- General problem Knowing the initial data and the morphisms are not C rþ1 ): infinitesimal evolution rule, what can we tell about the long-time evolution of the system? On very low dimensional spaces (diffeomorphisms of the circle and vector fields on compact surfaces) the The dynamics of a dynamical system (differential dynamics of generic systems (indeed in a open and equation or function) is the behavior of the orbits, dense subset of systems) is very simple (called Morse– when the time tends to infinity. The aim of Smale) and well understood; see the subsection ‘‘dynamical systems’’ is to produce a general ‘‘Genericpropertiesofthelow-dimensional procedure for describing the dynamics of any systems.’’ system. For example, Conley’s theory presented in Inhigherdimensions,for Cr-topo logy, r>1,one the next section organizes the global dymamics of a has generic and locally generic properties related general system using regions concentrating the orbit to the periodic orbits, like the Kupka–Smale accumulation and recurrence and splits these regions property(seethesubsection‘‘Kupka–Smaletheo- in elementary pieces: the chain recurrence classes. rem’’)andtheNewhousephenomenon(seethe We focus our study on C r-diffeomorphisms F (i.e., F subsection‘‘Local 2C-genericityofwildbehavior and F1 are r times continuously derivable) on a forsurfacediffeomorphisms’’).However,westill compact smooth manifold M (most of the notions and do not know if the dynamics of Cr-generic results presented here also hold for vector fields). Even diffeomorphisms is well approached by their for very regular systems (F algebraic) of a low- periodic orbits, so that one is still far from a dimensional space ( dim (M) = 2), the dynamics may global understanding of Cr-generic dynamics. be chaotic and very unstable: one cannot hope for a For the C1-topology, perturbation lemmas show that precise description of all systems. Furthermore, neither the global dynamics is very well approximated by the initial data of a concrete system nor the infinitesi- periodicorbits(seethesection‘‘C1-genericsystems: mal-evolution rule are known exactly: fragile proper- globaldynamicsandperiodicorbits’’).Onethen ties describe the evolution of the theoretical model, and dividesgenericsystemsin‘‘tame’’systems,witha not of the real system. For these reasons, we are mostly global dynamics analoguous to hyperbolic dynamics, interested in properties that are persistent, in some and ‘‘wild’’ systems, which present infinitely many sense, by small perturbations of the dynamical system. dynamically independent regions. The notion of The notion of small perturbations of the system dominatedsplitting(seethesection‘‘Hyperbolic requires a topology on the space Diffr(M)ofC r- propertiesof 1C-genericdiffeormorphisms’’)seems diffeomorphisms: two diffeomorphisms are close for to play an important role in this division. the C r-topology if all their partial derivatives of order r are close at each point of M. Endowed with this topology, Diff r(M) is a complete metric space. The open and dense subsets of Diffr(M)providethe Results on General Systems natural topological notion of ‘‘almost all’’ F. Genericity Notions of Recurrence is a weaker notion: by Baire’s theorem, if OTi, i 2 N,are Some regions of M are considered as the heart of the dense and open subsets, the intersection i2N Oi is a dense subset. A subset is called residual if it contains dynamics: such a countable intersection of dense open subsets. A Per(F) denotes the set of periodic points x 2 M of property P is generic if it is verified on a residual F, that is, Fn(x) = x for some n > 0. subset. By a practical abuse of language, one says: A point x is recurrent if its orbit comes back ‘‘C r-generic diffeomorphisms verify P’’ arbitrarily close to x, infinitely many times. Rec(F) denotes the set of recurrent points. A countable intersection of residual sets is a residual The limit set Lim(F) is the union of all the set. Hence, if {Pi}, i 2 N, is a countable family of accumulation points of all the orbits of F.

2 A point x is ‘‘wandering’’ if it admits a neighbor- with an associated Lyapunov function. Conley hood Ux M disjoint from all its iterates (1978) proved that n \ F (Ux), n > 0. The nonwandering set (F) is the set of the nonwandering points. RðFÞ¼ ðAi [ RiÞ R(F) is the set of chain recurrent points, that is, i2N points x 2 M which look like periodic points if we This induces a natural partition of R(F)inequiva- allow small mistakes at each iteration: for any lence classes: x y if x 2 Ai , y 2 Ai. Conley proved ">0, there is a sequence x = x0, x1, ..., xk = x that x y iff, for any ">0, there are "-pseudo orbits where d(f (xi), xiþ1) <" (such a sequence is an from x to y and vice versa. The equivalence classes "-pseudo-orbit). for are called chain recurrence classes. Now, considering an average of the Lyapunov A periodic point is recurrent, a recurrent point is a functions one gets the following result: there is a limit point, a limit point is nonwandering, and a i continuous function ’: M ! R with the following nonwandering point is chain recurrent: properties: PerðFÞRecðFÞLimðFÞðFÞRðFÞ ’(F(x)) ’(x) for every x 2 M, (i.e., ’ is a All these sets are invariant under F,and(F) and Lyapunov function); R(F) are compact subsets of M. There are diffeo- ’(F(x)) = ’(x) , x 2R(F); morphisms F for which the closures of these sets are for x, y 2R(F), ’(x) = ’(y) , x y;and distinct: the image ’(R(F) is a compact subset of R with empty interior. A rotation x 7! x þ with irrational angle 2 RnQ on the circle S1 = R=Z has no periodic This result is called the ‘‘fundamental theorem of points but every point is recurrent. dynamical systems’’ by several authors (see The map x 7! x þ (1=4)(1 þ cos (2x)) induces Robinson (1999)). on the circle S1 a diffeomorphism F having a Any orbit is ’-decreasing from a chain recurrence unique fixed point at x = 1=2; one verifies that class to another chain reccurence class (the global (F) = {1=2} and R(F) is the whole circle S1. dynamics of F looks like the dynamics of the gradient flow of a function , the chain recurrence An invariant compact set K M is transitive if there classes supplying the singularities of ). However, is x 2 K whose forward orbit is dense in K. Generic this description of the dynamics may be very rough: points x 2 K have their forward and backward if F preserves the volume, Poincare´’s recurrence orbits dense in K: in this sense, transitive sets are theorem implies that (F) = R(F) = M; the whole M dynamically indecomposable. is the unique chain recurrence class and the function ’ of Conley’s theorem is constant. Conley’s Theory: Pairs Attractor/Repeller and Conley’s theory provides a general procedure for Chain Recurrence Classes describing the global topological dynamics of a A trapping region U M is a compact set whose system: one has to characterize the chain recurrence image F(U) is contained in theT interior of U.By classes, the dynamics in restriction to each class, n definition, the intersection A = n0 F (U)isan the stable set of each class (i.e., the set of points attractor of F:anyorbitinU ‘‘goes to A.’’ Denote by whose positive orbits goes to the class), and the V the complement of the interior of U:itisatrappingT relative positions of these stable sets. 1 n region for F and the intersection R = n0 F (V)is a repeller. Each orbit either is contained in A [ R,or ‘‘goes from the repeller to the attractor.’’ More Hyperbolicity precisely, there is a smooth function : M ! [0, 1] Smale’s hyperbolic theory is the first attempt to give (called Lyapunov function) equal to 1 on R and 0 on A, a global vision of almost all dynamical systems. In and strictly decreasing on the other orbits: this section we give a very quick overview of this ðFðxÞÞ < ðxÞ for x2= A [ R theory. For further details, see Hyperbolic Dynami- cal Systems. So, the chain recurrent set is contained in A [ R. Any compact set contained in U and containing the Hyperbolic Periodic Orbits interior of F(U) is a trapping region inducing the same attracter and repeller pair (A,R); hence, the set A fixed point x of F is hyperbolic if the derivative of attracter/repeller pairs is countable. We denote by DF(x) has no (neither real nor complex) eigenvalue (Ai, Ri, i), i 2 N, the family of these pairs endowed with modulus equal to 1. The tangent space at x

3 s u s u splits as TxM = E E , where E and E are the orbits. However, for general systems, homoclinic DF(x)-invariant spaces corresponding to the eigen- classes are not necessarily disjoint. values of moduli <1 and >1, respectively. There are For more details, see Homoclinic Phenomena. Cr-injectively immersed F-invariant submanifolds Ws(x) and Wu(x) tangent at x to Es and Eu; the Smale’s Hyperbolic Theory stable manifold Ws(x) is the set of points y whose forward orbit goes to x. The implicit-function A diffeomorphism F is Morse–Smale if (F) = Per(F) theorem implies that a hyperbolic fixed point x is finite and hyperbolic, and if Ws(x) is tranverse to varies (locally) continuously with F; (compact parts Wu(y)foranyx, y 2 Per(F). Morse–Smale diffeo- of) the stable and unstable manifolds vary continu- morphisms have a very simple dynamics, similar to ously for the Cr-topology when F varies with the the one of the gradient flow of a Morse function; apart Cr-topology. from periodic points and invariant manifolds of A periodic point x of period n is hyperbolic if it is periodic saddles, each orbit goes from a source to a a hyperbolic fixed point of Fn and its invariant sink (hyperbolic periodic repellers and attractors). manifolds are the corresponding invariant manifolds Furthermore, Morse–Smale diffeomorphisms are for Fn. The stable and unstable manifold of the orbit C1-structurally stable, that is, any diffeomorphism s u 1 of x, Worb(x) and Worb(x), are the unions of the C -close to F is conjugated to F by a homeomorphism: invariant manifolds of the points in the orbit. the topological dynamics of F remains unchanged by small C1-perturbation. Morse–Smale vector fields Homoclinic Classes were known (Andronov and Pontryagin, 1937)to characterize the structural stability of vector fields on Distinct stable manifolds are always disjoint; how- 2 ever, stable and unstable manifolds may intersect. At the sphere S . However, a diffeomorphism having the end of the nineteenth century, Poincare´ noted transverse homoclinic intersections is robustly not Morse–Smale, so that Morse–Smale diffeomorphisms that the existence of transverse homoclinic orbits, r s are not C -dense, on any compact manifold of that is, transverse intersection of Worb(x) with u dimension 2. In the early 1960s, Smale generalized Worb(x) (other than the orbit of x), implies a very rich dynamical behavior: indeed, Birkhoff proved the notion of hyperbolicity for nonperiodic sets in that any transverse homoclinic point is accumulated order to get a model for homoclinic orbits. The goal of by a sequence of periodic orbits (see Figure 1). The the theory was to cover a whole dense open set of all homoclinic class H(x) of a periodic orbit is the dynamical systems. closure of the transverse homoclinic point associated An invariant compact set K is hyperbolic if the tangent space TMjK of M over K splits as the direct to x: s u sum TMK = E E of two DF-invariant vector s u bundles, where the vectors in Es and Eu are HðpÞ¼WorbðxÞ\WorbðxÞ uniformly contracted and expanded, respectively, There is an equivalent definition of the homoclinic by Fn,forsomen > 0. Hyperbolic sets persist class of x: we say that two hyperbolic periodic under small C1-perturbations of the dynamics: any s points x and y are homoclinically related if Worb(x) diffeomorphism G which is C1-close enough to F and Wu (x) intersect transversally Wu (y)and orb orb admits a hyperbolic compact set KG close to K and Ws (y), respectively; this defines an equivalence orb the restrictions of F and G to K and KG are relation in Perhyp(F) and the homoclinic classes are conjugated by a homeomorphism close to the the closure of the equivalence classes. identity. Hyperbolic compact sets have well- The homoclinic classes are transitive invariant defined invariant (stable and unstable) manifolds, compact sets canonically associated to the periodic tangent (at the points of K)toEs and Eu and the (local) invariant manifolds of KG vary locally continuously with G. The existence of hyperbolic sets is very common: x if y is a transverse homoclinic point associated to a hyperbolic periodic point x, then there is a transitive hyperbolic set containing x and y. Diffeomorphisms for which R(F) is hyperbolic are now well understood: the chain recurrence 2 f (x) f(x) classes are homoclinic classes, finitely many, and transitive, and admit a combinatorical model Figure 1 A transverse homoclinic orbit. (subshift of finite type). Some of them are

4 attractors or repellers, and the basins of the a nonperiodic point comes from a source and goes attractors cover a dense open subset of M. If, to a sink. Two Cr-generic diffeomorphisms of S1 are furthermore, all the stable and unstable manifolds conjugated iff they have same rotation number and of points in R(f ) are transverse, the diffeomorph- same number of periodic points. ism is C1-structurally stable (Robbin 1971, This simple behavior has been generalized in 1962 Robinson 1976); indeed, this condition, called by Peixoto for vector fields on compact orientable ‘‘axiom A þ strong transversality,’’ is equivalent surfaces S. Vector fields X in a Cr-dense and open to the C1-structural stability (Man˜e´ 1988). subset are Morse–Smale, hence structurally stable In 1970, Abraham and Smale built examples of (see Palis and de Melo (1982) for a detailed proof). robustly non-axiom A diffeomorphisms, when Peixoto gives a complete classification of these dim M 3: the dream of a global understanding vector fields, up to topological equivalence. of dynamical systems was postponed. However, Peixoto’s argument uses the fact that the return hyperbolicity remains a key tool in the study of maps of the vector field on transverse sections are dynamical systems, even for nonhyperbolic increasing functions: this helped control the effect systems. on the dynamics of small ‘‘monotonous’’ perturbations, and allowed him to destroy any nontrivial r recurrences. Peixoto’s result remains true on non- C -Generic Systems orientable surfaces for the C1-topology but remains Kupka–Smale Theorem an open question for r > 1: is the set of Morse– Smale vector fields C2-dense, for S nonorientable Thom’s transversality theorem asserts that two closed surface? submanifolds can always be put in tranverse posi- tion by a Cr-small perturbations. Hence, for F in an open and dense subset of Diffr(M), r 1, the graph Local C2-Genericity of Wild Behavior for Surface of F in M M is transverse to the diagonal Diffeomorphisms ={(x, x), x 2 M}: F has finitely many fixed points The generic systems we have seen above have a very xi, depending locally continuously on F, and 1 is not simple dynamics, simpler than the general systems. an eigenvalue of the differential DF(xi). Small local This is not always the case. In the 1970s, Newhouse perturbations in the neighborhood of the xi avoid exhibited a C2-open set ODiff2(S2) (where S2 eigenvalue of modulus equal to 1: one gets a dense r r denotes the two-dimensional sphere), such that and open subset O1 of Diff (M) such that every fixed C2-generic diffeomorphisms F 2O have infinitely point is hyperbolic. This argument, adapted for 2 r many hyperbolic periodic sinks. In fact, C -generic periodic points, provides a dense and open set On r diffeomorphisms in O present many other patholo- Diff (M), such that everyT periodic point of period n r gical properties: for instance, it has been recently is hyperbolic. Now n2N On is a residual subset of r noted that they have uncountably many chain Diff (M), for which every periodic point is recurrence classes without periodic orbits. Densely hyperbolic. (but not generically) in O, they present many other T Similarly, the set of diffeomorphisms F 2 n r phenomena, such as strange (Henon-like) attractors i = 0 Oi (M) such that all the disks of size n,of (see Lyapunov Exponents and Strange Attractors). invariant manifolds of periodic points of period less This phenomenon appears each time that a that n, are pairwise transverse, is open and dense. diffeomorphism F0 admits a hyperbolic periodic One gets the Kupka–Smale theorem (see Palis and de point x whose invariant manifolds Ws(x)and r Melo (1982) for a detailed exposition): for C -generic u s r W (x) are tangent at some point p 2 W (x) \ diffeomorphisms F 2 Diff (M), every periodic orbit is Wu(x)(p is a homoclinic tangency associated to x). s u hyperbolic and W (x) is transverse to W (y) for Homoclinic tangencies appear locally as a codimen- x, y 2 Per(F). sion-1 submanifold of Diff2(S2); they are such a simple phenomenon that they appear in very natural Generic Properties of Low-Dimensional Systems contexts. When a small perturbation transforms the Poincare´–Denjoy theory describes the topological tangency into tranverse intersections, a new hyper- dynamics of all diffeomorphisms of the circle S1 (see bolic set K with very large fractal dimensions is Homeomorphisms and Diffeomorphisms of the created. The local stable and unstable manifolds of Circle). Diffeomorphisms in an open and dense K, each homeomorphic to the product of a Cantor r 1 2 subset of Diffþ(S ) have a nonempty finite set of set by a segment, present tangencies in a C -robust way, periodic orbits, all hyperbolic, and alternately that is, for F in some C2-open set O (see Figure 2). attracting (sink) or repelling (source). The orbit of As a consequence, for a C2-dense subset of O, the

5 2 For the C topology, thep distancesffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d(xi, x0) need to remain greater than d(xn, x0)=" d(xn, x0). This new difficulty is why the C2-closing lemma remains an open question. Pugh’s argument does not suffice to create homoclinic point for a periodic orbit whose unstable manifold accumulates on the stable one. In 1998, Hayashi solved this problem proving the Connecting lemma (Hayashi 1997) Let y and z be two points such that the forward orbit of y and the Figure 2 Robust tangencies. backward orbit of z accumulate on the same nonperiodic point x. Fix some ">0. There is N > invariant manifolds of the point x present some 0 and a "-C1-perturbation G of F such that Gn(y) = z tangency (this is not generic, by Kupka–Smale for some n > 0, and G F out of an arbitrary small theorem). If the Jacobian of F at x is <1, each neighborhood of {x, F(x), ..., FN(x)}. tangency allows to create one more sink, by an arbitrarily small perturbation. Hence, the sets of Using Hayashi’s arguments, we (with Crovisier) diffeomorphisms having more than n hyperbolic proved the following lemma: sinks are dense open subsets of O, and the Connecting lemma for pseudo-orbits (Bonatti and intersection of all these dense open subsets is the Crovisier 2004) Assume that all periodic orbits of F announced residual set. See Palis and Takens are hyperbolic; consider x, y 2 M such that, for any (1993) for details on this deep argument. ">0, there are "-pseudo-orbits joining x to y; then there are arbitrarily small C1-perturbations of F for which the positive orbit of x passes through y. C1-Generic Systems: Global Dynamics and Periodic Orbits Densities of Periodic Orbits See Bonatti et al. (2004), Chapter 10 and Appendix A, As a consequence of the perturbations lemma above, for a more detailed exposition and precise we (Bonatti and Crovisier 2004) proved that for 1 references. FC-generic,

RðFÞ¼ðFÞ¼PerhypðFÞ Perturbations of Orbits: Closing and Connecting where Per (F) denotes the closure of the set of Lemmas hyp hyperbolic periodic points. In 1968, Pugh proved the following Lemma. For this, consider the map : F 7! (F) = Perhyp(F) defined on Diff1(M) and with value in K(M), space Closing lemma If x is a nonwandering point of a of all compact subsets of M, endowed with the diffeomorphism F, then there are diffeomorphisms Hausdorff topology. Per (F) may be approximated G arbitrarily C1-close to F, such that x is periodic hyp by a finite set of hyperbolic periodic points, and this for G. set varies continuously with F; so Perhyp(F) varies n Consider a segment x0, ..., xn = F (x0) of orbit lower-semicontinuously with F: for G very close to such that xn is very close to x0 = x; one would like F, Perhyp(G) cannot be very much smaller than to take G close to F such that G(xn) = x0, and Perhyp(F). As a consequence, a result from general 1 G(xi) = F(xi) = xiþ1 for i 6¼ n. This idea works for topology asserts that, for C -generic F, the map is the C0-topology (so that the C0-closing lemma is continuous at F. On the other hand, C1-generic easy). However, if one wants G "-C1-close to F, one diffeomorphisms are Kupka–Smale, so that the needs that the points xi, i 2 {1, ..., n 1}, remain at connecting lemma for pseudo-orbits may apply: distance d(xi, x0) greater than C(d(xn, x0)="), where if x 2R(F), x can be turned into a hyperbolic C bounds kDf k on M.IfC=" is very large, such a periodic point by a C1-small perturbation of F. So, segment of orbit does not exist. Pugh solved this if x 2= Perhyp(F), F is not a continuity point of , difficulty in two steps: the perturbation is first leading to a contradiction. spread along a segment of orbit of x in order to Furthermore, Crovisier proved the following 1 decrease this constant; then a subsegment y0, ..., yk result: ‘‘for C -generic diffeomorphisms, each chain of x0, ..., xn is selected, verifying the geometrical recurrence class is the limit, for the Hausdorff condition. distance, of a sequence of periodic orbits.’’

6 This good approximation of the global dynamics A local version (in the neighborhood of a chain by the periodic orbits will now allow us to recurrence class) of this argument shows that, for better understand the chain recurrence classes of C1-generic diffeomorphisms, any isolated chain C1-generic diffeomorphisms. recurrence classe C is robustly isolated: for any diffeomorphism G, C1-close enough to F, the Chain Recurrence Classes/Homoclinic Classes intersection of R(G) with a small neighborhood of 1 of C -Generic Systems C is a unique chain recurrence class CG close to C. One says that a diffeomorphism is ‘‘tame’’ if each Tranverse intersections of invariant manifolds of chain recurrence class is robustly isolated. We hyperbolic orbits are robust and vary locally denote by T (M) Diff1(M) the (C1-open) set of continuously with the diffeomorphisms F. So, the tame diffeomorphisms and by W(M) the comple- homoclinic class H(x) of a periodic point x varies ment of the closure of T (M). C1-generic diffeo- lower-semicontinuously with F (on the open set morphisms in W(M) have infinitely many disjoint where the continuation of x is defined). As a homoclinic classes, and are called ‘‘wild’’ consequence, for Cr-generic diffeomorphisms (r diffeomorphisms. 1), each homoclinic class varies continuously with F. Generic tame diffeomorphisms have a global Using the connecting lemma, Arnaud (2001) proved dynamics analogous to hyperbolic systems: the the following result: ‘‘for Kupka–Smale diffeo- chain recurrence set admits a partition into finitely morphisms, if the closures Wu (x)andWs (x) orb orb many homoclinic classes varying continuously with have some intersection point z, then a C1-pertuba- the dynamics. Every point belongs to the stable set tion of F creates a tranverse intersection of Wu (x) orb of one of these classes. Some of the homoclinic and Ws (x)atz.’’ So, if z2= H(x), then F is not a orb classes are (transitive) topological attractors, and the continuity point of the function F 7! H(x, F). Hence, union of the basins covers a dense open subset of M, for C1-generic diffeomorphisms F and for every and the basins vary continuously with F (Carballo periodic point x, Morales 2003). It remains to get a good description u s of the dynamics in the homoclinic classes, and HðxÞ¼WorbðxÞ\WorbðxÞ particularly in the attractors. As we shall see in the u s In the same way, Worb(x) and Worb(x) vary locally next section, tame behavior requires some kind of lower-semicontinuously with F so that, for F weak hyperbolicity. Indeed, in dimension 2, tame r C -generic, the closures of the invariant manifolds diffeomorphisms satisfy axiom A and the noncycle of each periodic point vary locally continuously. For condition. Kupka–Smale diffeomorphisms, the connecting As of now, very little is known about wild lemma for pseudo-orbits implies: ‘‘if z is a point in systems. One knows some semilocal mechanisms the chain recurrence class of a periodic point x, then generating locally C1-generic wild dynamics, there- 1 a C -small perturbation of F puts z on the unstable fore proving their existence on any manifold with u manifold of x’’; so, if z 2= Worb(x), then F is not a dimension dim (M) 3 (the existence of wild diffeo- u 1 continuity point of the function F 7! Worb(x, F). morphisms in dimension 2, for the C -topology, 1 Hence, for C -generic diffeomorphisms F and for remains an open problem). Some of the known every periodic point x, the chain recurrence class of examples exhibit a universal dynamics: they admit u s x is contained in Worb(x) \ Worb(x), and, therefore, infinitely many disjoint periodic disks such that, up coincides with the homoclinic class of x. This to renormalization, the return maps on these disks argument proves: induce a dense subset of diffeomorphisms of the For a C1-generic diffeomorphism F, each homoclinic class disk. Hence, these locally generic diffeomorphisms H(x) is a chain recurrence class of F (of Conley’s theory): present infinitely many times any robust property of a chain recurrence class containing a periodic point x diffeomorphisms of the disk. coincides with the homoclinic class H(x). In particular, two homoclinic classes are either disjoint or equal. Ergodic Properties A point x is well closable if, for any ">0 there is Tame and Wild Systems G "-C1-close to F such that x is periodic for G and d(Fi(x), Gi(x)) <" for i 2 {0, ..., p}, p being the For generic diffeomorphisms, the number N(F) 2 period of x. As an important refinement of Pugh’s N [ {1} of homoclinic classes varies lower-semicon- closing lemma, Man˜e´ proved the following lemma: tinuously with F. One deduces that N(F)islocally constant on a residual subset of Diff1(M)(Abdenur Ergodic closing lemma For any F-invariant prob- 2003). ability, almost every point is well closable.

7 As a consequence, ‘‘for C1-generic diffeomoph- F-invariant set such that the tangent space of M at isms, any ergodic measure is the weak limit of a the points x 2 X admits a DF-invariant splitting sequence of Dirac measures on periodic orbits, Tx(M) = E1(x) Ek(x), the dimensions dim (Ei(x)) which converges also in the Hausdorff distance to being independent of x. This splitting is dominated if the support of .’’ the vectors in Eiþ1 are uniformly more expanded than It remains an open problem to know if, for the vectors in Ei: there exists ‘>0 such that, for C1-generic diffeomorphisms, the ergodic measures any x 2 X,anyi 2 {1, ..., k 1} and any unit vectors supported in a homoclinic class are approached by u 2 Ei(x)andv 2 Eiþ1(x), one has periodic orbits in this homoclinic class. ‘ 1 ‘ kDF ðuÞk < 2 kDF ðvÞk Conservative Systems Dominated splittings are always continuous, The connecting lemma for pseudo-orbits has been extend to the closure of X, and persist and vary adapted for volume preserving and symplectic continuously under C1-perturbation of F. diffeomorphisms, replacing the condition on the periodic orbits by another generic condition on the eigenvalues. As a consequence, one gets: ‘‘C1-generic Dominated Splittings versus Wild Behavior volume-preserving or symplectic diffeomorphisms Let { } be a set of hyperbolic periodic orbits. On are transitive, and M is a unique homoclinic class.’’ Si X = one considers the natural splitting Notice that the KAM theory implies that this i TMj = Es Eu induced by the hyperbolicity of the result is wrong for C4-generic diffeomorphisms, the X . Man˜e´ (1982) proved: ‘‘if there is a C1-neighbor- persistence of invariant tori allowing to break i hood of F on which each remains hyperbolic, then robustly the transitivity. i the splitting TMj = Es Eu is dominated.’’ The Oxtoby–Ulam (1941) theorem asserts that X A generalization of Man˜e´’s result shows: ‘‘if a C0-generic volume-preserving homeomorphisms are homoclinic class H(x) has no dominated splitting, ergodic. The ergodicity of C1-generic volume- then for any ">0 there is a periodic orbit in H(x) preserving diffeomorphisms remains an open question. whose derivative at the period can be turned into an homothety, by an "-small perturbation of the Hyperbolic Properties of C1-Generic derivative of F along the points of ’’; in particular, Diffeomorphisms this periodic orbit can be turned into a sink or a source. As a consequence, one gets: ‘‘for C1-generic For a more detailed exposition of hyperbolic proper- diffeomorphisms F, any homoclinic class either has a 1 ties of C -generic diffeomorphisms, the reader is dominated splitting or is contained in the closure of referred to Bonatti et al. (2004, chapter 7 and the (infinite) set of sinks and sources.’’ appendix B). This argument has been used in two directions:

Perturbations of Products of Matrices Tame systems must satisfy some hyperbolicity. In

1 fact, using the ergodic closing lemma, one proves The C -topology enables us to do small perturbations of that the homoclinic classes H(x) of tame diffeo- the differential DF at a point x without perturbing either morphisms are volume hyperbolic, that is, there is F(x)orF out of an arbitrarily small neighborhood of x. a dominated splitting TM = E1 Ek over Hence, one can perturb the differential of F along a H(x) such that DF contracts uniformly the periodic orbit, without changing this periodic orbit volume in E1 and expands uniformly the volume (Frank’s lemma). When x is a periodic point of period n, n in Ek. the differential of F at x is fundamental for knowing the If F admits a homoclinic class H(x) which is local behavior of the dynamics. This differential is (up to robustly without dominated splittings, then gen- a choice of local coordinates) a product of the matrices i eric diffeomorphisms in the neighborhood of F are DF(xi), where xi = F (x). So, the control of the wild: at this time this is the unique known way to dynamical effect of local perturbations along a periodic get wild systems. orbit comes from a problem of linear algebra: ‘‘consider a product A = An An1 A1 of n 0 bounded d See also: Cellular Automata; Chaos and Attractors; linear ismorphisms of R ; how do the eigenvalues and Fractal Dimensions in Dynamics; Homeomorphisms and the eigenspaces of A vary under small perturbations of Diffeomorphisms of the Circle; Homoclinic Phenomena; the Ai?’’ Hyperbolic Dynamical Systems; Lyapunov Exponents A partial answer to this general problem uses and Strange Attractors; Polygonal Billiards; Singularity the notion of dominated splitting. Let X M be an and Bifurcation Theory; Synchronization of Chaos.

8 Further Reading Hayashi S (1997) Connecting invariant manifolds and the solution of the C1 stability and -stability conjectures for Abdenur F (2003) Generic robustness of spectral decompositions. flows. Annals of Mathematics 145: 81–137. Annales Scientifiques de l’Ecole Normale Superieure IV 36(2): Man˜e´ R (1988) A proof of the C1 stability conjecture. Inst. 213–224. Hautes E´ tudes Sci. Publ. Math. 66: 161–210. Andronov A and Pontryagin L (1937) Syste`mes grossiers. Dokl. Palis J and de Melo W (1982) Gemetric Theory of Dynamical Akad. Nauk. USSR 14: 247–251. Systems, An Introduction. New York–Berlin: Springer. Arnaud M-C (2001) Creátion de connexions en topologie C1. Palis J and Takens F (1993) Hyperbolicity and Sensitive-Chaotic Ergodic Theory and Dynamical Systems 21(2): 339–381. Dynamics at Homoclinic Bifurcation. Cambridge: Cambridge Bonatti C and Crovisier S (2004) Rećurrence et geńe´ricite´. University Press. Inventiones Mathematical 158(1): 33–104 (French) (see also Robbin JW (1971) A structural stability theorem. Annals of the short English version: (2003) Recurrence and genericity. Mathematics 94(2): 447–493. Comptes Rendus Mathematique de l’Academie des Sciences Robinson C (1974) Structural stability of vector fields. Annals of Paris 336 (10): 839–844). Mathematics 99(2): 154–175 (errata. Annals of Mathematics Bonatti C, Dıàz LJ, and Viana M (2004) Dynamics Beyond 101(2): 368 (1975)). Uniform Hyperbolicity. Encyclopedia of Mathematical Robinson C (1976) Structural stability of C1 diffeomorphisms. Sciences, vol. 102. Berlin: Springer. Journal of Differential Equations 22(1): 28–73. Carballo CM and Morales CA (2003) Homoclinic classes and Robinson C (1999) Dynamical Systems. Studies in Advanced finitude of attractors for vector fields on n-manifolds. Mathematics. Boca Raton, FL: CRC Press. Bulletins of the London Mathematical Society 35(1): 85–91.