aa r X i v : . [ m a t h . D S ] A p r HYPERBOLIC DYNAMICAL SYSTEMS
VITOR ARA ´UJO AND MARCELO VIANA C ONTENTS
Glossary 1Definition 21. Introduction 32. Linear systems 33. Local theory 54. Hyperbolic behavior: examples 65. Hyperbolic sets 86. Uniformly hyperbolic systems 117. Attractors and physical measures 128. Obstructions to hyperbolicity 149. Partial hyperbolicity 1510. Non-uniform hyperbolicity - Linear theory 1711. Non-uniformly hyperbolic systems 1912. Future directions 21References 21G
LOSSARY
Homeomorphism, diffeomorphism. A homeomorphism is a continuous map f : M → N which is one-to-one and onto, and whose inverse f − : N → M is also continuous. It maybe seen as a global continuous change of coordinates. We call f a diffeomorphism if, inaddition, both it and its inverse are smooth. When M = N , the iterated n -fold composition f ◦ n . . . ◦ f is denoted by f n . By convention, f is the identity map, and f − n = ( f n ) − =( f − ) n for n ≥ Smooth flow.
A flow f t : M → M is a family of diffeomorphisms depending in a smoothfashion on a parameter t ∈ R and satisfying f s + t = f s ◦ f t for all s , t ∈ R . This prop-erty implies that f is the identity map. Flows usually arise as solutions of autonomousdifferential equations: let t f t ( v ) denote the solution of˙ X = F ( X ) , X ( ) = v (1)and assume solutions are defined for all times; then the family f t thus defined is a flow (atleast as smooth as the vector field F itself). The vector field may be recovered from theflow, through the relation F ( X ) = ¶ t f t ( X ) | t = . Date : January 2, 2014. C k topology. Two maps admitting continuous derivatives are said to be C -close if theyare uniformly close, and so are their derivatives. More generally, given any k ≥
1, we saythat two maps are C k -close if they admit continuous derivatives up to order k , and theirderivatives of order i are uniformly close, for every i = , , . . . , k . This defines a topologyin the space of maps of class C k . Foliation.
A foliation is a partition of a subset of the ambient space into smooth sub-manifolds, that one calls leaves of the foliation, all with the same dimension and varyingcontinuously from one point to the other. For instance, the trajectories of a vector field F ,that is, the solutions of equation (1), form a 1-dimensional foliation (the leaves are curves)of the complement of the set of zeros of F . The main examples of foliations in the contextof this work are the families of stable and unstable manifolds of hyperbolic sets. Attractor.
A subset L of the ambient space M is invariant under a transformation f if f − ( L ) = L , that is, a point is in L if and only if its image is. L is invariant under a flow ifit is invariant under f t for all t ∈ R . An attractor is a compact invariant subset L such thatthe trajectories of all points in a neighborhood U converge to L as times goes to infinity,and L is dynamically indecomposable (or transitive ): there is some trajectory dense in L .Sometimes one asks convergence only for points in some “large” subset of a neighborhood U of L , and dynamical indecomposability can also be defined in somewhat different ways.However, the formulations we just gave are fine in the uniformly hyperbolic context. Limit sets.
The w -limit set of a trajectory f n ( x ) , n ∈ Z is the set w ( x ) of all accumulationpoints of the trajectory as time n goes to + ¥ . The a -limit set is defined analogously,with n → − ¥ . The corresponding notions for continuous time systems (flows) are definedanalogously. The limit set L ( f ) (or L ( f t ) , in the flow case) is the closure of the union of all’ w -limit and all a -limit sets. The non-wandering set W ( f ) (or W ( f t ) , in the flow case) isthat set of points such that every neighborhood U contains some point whose orbit returnsto U in future time (then some point returns to U in past time as well). When the ambientspace is compact all these sets are non-empty. Moreover,the limit set is contained in thenon-wandering set. Invariant measure.
A probability measure µ in the ambient space M is invariant under atransformation f if µ ( f − ( A )) = µ ( A ) for all measurable subsets A . This means that the“events” x ∈ A and f ( x ) ∈ A have equally probable. We say µ is invariant under a flowif it is invariant under f t for all t . An invariant probability measure µ is ergodic if everyinvariant set A has either zero or full measure. An equivalently condition is that µ can notbe decomposed as a convex combination of invariant probability measures, that is, one cannot have µ = aµ + ( − a ) µ with 0 < a < µ , µ invariant.D EFINITION
In general terms, a smooth dynamical system is called hyperbolic if the tangent spaceover the asymptotic part of the phase space splits into two complementary directions, onewhich is contracted and the other which is expanded under the action of the system. In theclassical, so-called uniformly hyperbolic case, the asymptotic part of the phase space isembodied by the limit set and, most crucially, one requires the expansion and contractionrates to be uniform. Uniformly hyperbolic systems are now fairly well understood. Theymay exhibit very complex behavior which, nevertheless, admits a very precise descrip-tion. Moreover, uniform hyperbolicity is the main ingredient for characterizing structural
YPERBOLIC DYNAMICAL SYSTEMS 3 stability of a dynamical system. Over the years the notion of hyperbolicity was broad-ened (non-uniform hyperbolicity) and relaxed (partial hyperbolicity, dominated splitting)to encompass a much larger class of systems, and has become a paradigm for complexdynamcial evolution. 1. I
NTRODUCTION
The theory of uniformly hyperbolic dynamical systems was initiated in the 1960’s(though its roots stretch far back into the 19th century) by S. Smale, his students and col-laborators, in the west, and D. Anosov, Ya. Sinai, V. Arnold, in the former Soviet Union.It came to encompass a detailed description of a large class of systems, often with verycomplex evolution. Moreover, it provided a very precise characterization of structurallystable dynamics, which was one of its original main goals.The early developments were motivated by the problem of characterizing structural sta-bility of dynamical systems, a notion that had been introduced in the 1930’s by A. An-dronov and L. Pontryagin. Inspired by the pioneering work of M. Peixoto on circle mapsand surface flows, Smale introduced a class of gradient-like systems, having a finite num-ber of periodic orbits, which should be structurally stable and, moreover, should constitutethe majority (an open and dense subset) of all dynamical systems. Stability and opennesswere eventually established, in the thesis of J. Palis. However, contemporary results of M.Levinson, based on previous work by M. Cartwright and J. Littlewood, provided examplesof open subsets of dynamical systems all of which have an infinite number of periodicorbits.In order to try and understand such phenomenon, Smale introduced a simple geometricmodel, the now famous ”horseshoe map”, for which infinitely many periodic orbits existin a robust way. Another important example of structurally stable system which is notgradient like was R. Thom’s so-called ”cat map”. The crucial common feature of thesemodels is hyperbolicity: the tangent space at each point splits into two complementardirections such that the derivative contracts one of these directions and expands the other,at uniform rates.In global terms, a dynamical system is called uniformly hyperbolic , or Axiom A, ifits limit set has this hyperbolicity property we have just described. The mathematicaltheory of such systems, which is the main topic of this paper, is now well developped andconstitutes a main paradigm for the behavior of ”chaotic” systems. In our presentationwe go from local aspects (linear systems, local behavior, specific examples) to the globaltheory (hyperbolic sets, stability, ergodic theory). In the final sections we discuss severalimportant extensions (strange attractors, partial hyperbolicity, non-uniform hyperbolicity)that have much broadened the scope of the theory.2. L
INEAR SYSTEMS
Let us start by introducing the phenomenon of hyperbolicity in the simplest possiblesetting, that of linear transformations and linear flows. Most of what we are going to sayapplies to both discrete time and continuous time systems in a fairly analogous way, andso at each point we refer to either one setting or the other. In depth presentations can befound in e.g. [8] and [6].The general solution of a system of linear ordinary differential equations˙ X = AX , X ( ) = v VITOR ARA ´UJO AND MARCELO VIANA where A is a constant n × n real matrix and v ∈ R n is fixed, is given by X ( t ) = e tA · v , t ∈ R , where e tA = (cid:229) ¥ n = ( tA ) n / n !. The linear flow is called hyperbolic if A has no eigenvalues onthe imaginary axis. Then the exponential matrix e A has no eigenvalues with norm 1. Thisproperty is very important for a number of reasons. Stable and unstable spaces.
For one thing it implies that all solutions have well-definedasymptotic behavior: they either converge to zero or diverge to infinity as time t goes to ± ¥ . More precisely, let • E s ( stable subspace ) be the subspace of R n spanned by the generalized eigenvectorassociated to eigenvalues of A with negative real part. • E u ( unstable subspace ) be the subspace of R n spanned by the generalized eigen-vector associated to eigenvalues of A with positive real partThen these subspaces are complementary, meaning that R n = E s ⊕ E u , and every solution e tA · v with v E s ∪ E u diverges to infinity both in the future and in the past. The solutionswith v ∈ E s converge to zero as t → + ¥ and go to infinity as t → − ¥ , and analogouslywhen v ∈ E u , reversing the direction of time. Robustness and density.
Another crucial feature of hyperbolicity is robustness : any ma-trix that is close to a hyperbolic one, in the sense that corresponding coefficients are close,is also hyperbolic. The stable and unstable subspaces need not coincide, of course, but thedimensions remain the same. In addition, hyperbolicity if dense : any matrix is close to ahyperbolic one. That is because, up to arbitrarily small modifications of the coefficients,one may force all eigenvalues to move out of the imaginary axis.
Stability, index of a fixed point.
In addition to robustness, hyperbolicity also implies stability : if B is close to a hyperbolic matrix A , in the sense we have just described, thenthe solutions of ˙ X = BX have essentially the same behavior as the solutions of ˙ X = AX .What we mean by “essentially the same behavior” is that there exists a global continuouschange of coordinates, that is, a homeomorphism h : R n → R n , that maps solutions of onesystem to solutions of the other, preserving the time parametrization: h (cid:0) e tA · v (cid:1) = e tB · h ( v ) for all t ∈ R . More generally, two hyperbolic linear flows are conjugated by a homeomorphism h if andonly if they have the same index , that is, the same number of eigenvalues with negative realpart. In general, h can not be taken to be a diffeomorphism: this is possible if and only ifthe two matrices A and B are obtained from one another via a change of basis. Notice thatin this case they must have the same eigenvalues, with the same multiplicities. Hyperbolic linear flows.
There is a corresponding notion of hiperbolicity for discretetime linear systems X n + = CX n , X = v with C a n × n real matrix. Namely, we say the system is hyperbolic if C has no eigenvaluein the unit circle. Thus a matrix A is hyperbolic in the sense of continuous time systems ifand only if its exponential C = e A is hyperbolic in the sense of discrete time systems. Theprevious observations (well-defined behavior, robustness, denseness and stability) remaintrue in discrete time. Two hyperbolic matrices are conjugate by a homeomorphism if andonly if they have the same index, that is, the same number of eigenvalues with norm lessthan 1, and they both either preserve or reverse orientation. YPERBOLIC DYNAMICAL SYSTEMS 5
3. L
OCAL THEORY
Now we move on to discuss the behavior of non-linear systems close to fixed or, moregenerally, periodic trajectories. By non-linear system we understand the iteration of adiffeomorphism f , or the evolution of a smooth flow f t , on some manifold M . The generalphilosophy is that the behavior of the system close to a hyperbolic fixed point very muchresembles the dynamics of its linear part.A fixed point p ∈ M of a diffeomorphism f : M → M is called hyperbolic if the linearpart D f p : T p M → T p M is a hyperbolic linear map, that is, if D f p has no eigenvalue withnorm 1. Similarly, an equilibrium point p of a smooth vector field F is hyperbolic if thederivative DF ( p ) has no pure imaginary eigenvalues. Hartman-Grobman theorem.
This theorem asserts that if p is a hyperbolic fixed pointof f : M → M then there are neighborhoods U of p in M and V of 0 in the tangent space T p M such that we can find a homeomorphism h : U → V such that h ◦ f = D f p ◦ h whenever the composition is defined. This property means that h maps orbits of D f ( p ) close to zero to orbits of f close to p . We say that h is a (local) conjugacy between thenon-linear system f and its linear part D f p . There is a corresponding similar theorem forflows near a hyperbolic equilibrium. In either case, in general h can not be taken to be adiffeomorphism. Stable sets.
The stable set of the hyperbolic fixed point p is defined by W s ( p ) = { x ∈ M : d ( f n ( x ) , f n ( p )) −−−−→ n → + ¥ } Given b > local stable set of size b >
0, defined by W s b ( p ) = { x ∈ M : d ( f n ( x ) , f n ( p )) ≤ b for all n ≥ } . The image of W s b under the conjugacy h is a neighborhood of the origin inside E s . It followsthat the local stable set is an embedded topological disk, with the same dimension as E s .Moreover, the orbits of the points in W s b ( p ) actually converges to the fixed point as timegoes to infinity. Therefore, z ∈ W s ( p ) ⇔ f n ( z ) ∈ W s b ( p ) for some n ≥ . Stable manifold theorem.
The stable manifold theorem asserts that W s b ( p ) is actually asmooth embedded disk, with the same order of differentiability as f itself, and it is tangentto E s at the point p . It follows that W s ( p ) is a smooth submanifold, injectively immersedin M . In general, W s ( p ) is not embedded in M : in many cases it has self-accumulationpoints. For these reasons one also refers to W s ( p ) and W s b ( p ) as stable manifolds of p .Unstable manifolds are defined analogously, replacing the transformation by its inverse. Local stability.
We call index of a diffeomorphism f at a hyperbolic fixed point p theindex of the linear part, that is, the number of eigenvalues of D f p with negative real part.By the Hartman-Grobman theorem and previous comments on linear systems, two dif-feomorphisms are locally conjugate near hyperbolic fixed points if and only if the stableindices and they both preserve/reverse orientation. In other words, the index together withthe sign of the Jacobian determinant form a complete set of invariants for local topologicalconjugacy.Let g be any diffeomorphism C -close to f . Then g has a unique fixed point p g close to p , and this fixed point is still hyperbolic. Moreover, the stable indices and the orientations VITOR ARA ´UJO AND MARCELO VIANA of the two diffeomorphisms at the corresponding fixed points coincide, and so they arelocally conjugate. This is called local stability near of diffeomorphisms hyperbolic fixedpoints. The same kind of result holds for flows near hyperbolic equilibria.4. H
YPERBOLIC BEHAVIOR : EXAMPLES
Now let us review some key examples of (semi)global hyperbolic dynamics. Thoroughdescriptions are available in e.g. [8], [6] and [9].
A linear torus automorphism.
Consider the linear transformation A : R → R given bythe following matrix, relative to the canonical base of the plane: (cid:18) (cid:19) . The 2-dimensional torus T is the quotient R / Z of the plane by the equivalence relation ( x , y ) ∼ ( x , y ) ⇔ ( x − x , y − y ) ∈ Z . Since A preserves the lattice Z of integer vectors, that is, since A ( Z ) = Z , the lineartransformation defines an invertible map f A : T → T in the quotient space, which is anexample of linear automorphism of T . We call affine line in T the projection under thequotient map of any affine line in the plane.The linear transformation A is hyperbolic, with eigenvalues 0 < l < < l , and thecorresponding eigenspaces E and E have irrational slope. For each point z ∈ T , let W i ( z ) denote the affine line through z and having the direction of E i , for i =
1, 2: • distances along W ( z ) are multiplied by l < f A • distances along W ( z ) are multiplied by 1 / l < f A .Thus we call W ( z ) stable manifold and W ( z ) unstable manifold of z (notice we are notassuming z to be periodic). Since the slopes are irrational, stable and unstable manifoldsare dense in the whole torus. From this fact one can deduce that the periodic points of f A form a dense subset of the torus, and that there exist points whose trajectories are dense in T . The latter property is called transitivity .An important feature of this systems is that its behavior is (globally) stable under smallperturbations: given any diffeomorphism g : T → T sufficiently C -close to f A , thereexists a homeomorphism h : T → T such that h ◦ g = f A ◦ h . In particular, g is alsotransitive and its periodic points form a dense subset of T . The Smale horseshoe.
Consider a stadium shaped region D in the plane divided into threesubregions, as depicted in Figure 1: two half disks, A and C , and a square, B . Next, considera map f : D → D mapping D back inside itself as described in Figure 1: the intersectionbetween B and f ( B ) consists of two rectangles, R and R , and f is affine on the pre-image of these rectangles, contracting the horizontal direction and expanding the verticaldirection.The set L = ∩ n ∈ Z f n ( B ) , formed by all the points whose orbits never leave the square B is totally disconnected, in fact, it is the product of two Cantor sets. A description of thedynamics on L may be obtained through the following coding of orbits. For each point z ∈ L and every time n ∈ Z the iterate f n ( z ) must belong to either R or R . We call itinerary of z the sequence { s n } n ∈ Z with values in the set { , } defined by f n ( z ) ∈ R s n forall n ∈ Z . The itinerary map L → { , } Z , z
7→ { s n } n ∈ Z YPERBOLIC DYNAMICAL SYSTEMS 7 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
ABCD f ( A ) f ( B ) f ( C ) IGURE
1. Horseshoe mapis a homeomorphism, and conjugates f restricted to L to the so-called shift map defined onthe space of sequences by { , } Z → { , } Z , { s n } n ∈ Z
7→ { s n + } n ∈ Z . Since the shift map is transitive, and its periodic points form a dense subset of the domain,it follows that the same is true for the horseshoe map on L .From the definition of f we get that distances along horizontal line segments throughpoints of L are contracted at a uniform rate under forward iteration and, dually, distancesalong vertical line segments through points of L are contracted at a uniform rate underbackward iteration. Thus, horizontal line segments are local stable sets and vertical linesegments are local unstable sets for the points of L .A striking feature of this system is the stability of its dynamics: given any diffeomor-phism g sufficiently C -close to f , its restriction to the set L g = ∩ n ∈ Z g n ( B ) is conjugateto the restriction of f to the set L = L f (and, consequently, is conjugate to the shift map).In addition, each point of L g has local stable and unstable sets which are smooth curvesegments, respectively, approximately horizontal and approximately vertical. The solenoid attractor.
The solid torus is the product space SS × D , where SS = R / Z is the circle and D = { z ∈ C : | z | < } is the unit disk in the complex plane. Consider themap f : SS × D → SS × D given by ( q , z ) ( q , a z + b e i q / ) , q ∈ R / Z and a , b ∈ R with a + b <
1. The latter condition ensures that the image f ( SS × D ) is strictly contained in SS × D . Geometrically, the image is a long thin domaingoing around the solid torus twice, as described in Figure 2. Then, for any n ≥
1, the cor-responding iterate f n ( SS × D ) is an increasingly thinner and longer domain that winds 2 k times around SS × D . The maximal invariant set L = ∩ n ≥ f n ( SS × D ) is called solenoid attractor . Notice that the forward orbit under f of every point in SS × D accumulates on L . One can also check that the restriction of f to the attractor is transitive,and the set of periodic points of f is dense in L . VITOR ARA ´UJO AND MARCELO VIANA SS × D f ( SS × D ) { q } × D F IGURE
2. The solenoid attractorIn addition L has a dense subset of periodic orbits and also a dense orbit. Moreoverevery point in a neighborhood of L converges to L and this is why this set is called an attractor . 5. H YPERBOLIC SETS
The notion we are now going to introduce distillates the crucial feature common to theexamples presented previously. A detailed presentation is given in e.g. [8] and [10]. Let f : M → M be a diffeomorphism on a manifold M . A compact invariant set L ⊂ M is a hyperbolic set for f if the tangent bundle over L admits a decomposition T L M = E u ⊕ E s , invariant under the derivative and such that k D f − | E u k < l and k D f | E s k < l for someconstant l < E s the stable bundleand E u the unstable bundle of f on the set L .The definition of hyperbolicity for an invariant set of a smooth flow containing no equi-libria is similar, except that one asks for an invariant decomposition T L M = E u ⊕ E ⊕ E s ,where E u and E s are as before and E is a line bundle tangent to the flow lines. An invariantset that contains equilibria is hyperbolic if and only it consists of a finite number of points,all of them hyperbolic equilibria. Cone fields.
The definition of hyperbolic set is difficult to use in concrete situations, be-cause, in most cases, one does not know the stable and unstable bundles explicitly. For-tunately, to prove that an invariant set is hyperbolic it suffices to have some approximateknowledge of these invariant subbundles. That is the contents of the invariant cone fieldcriterion: a compact invariant set is hyperbolic if and only if there exists some continu-ous (not necessarily invariant) decomposition T L M = E ⊕ E of the tangent bundle, someconstant l <
1, and some cone field around E C a ( x ) = { v = v + v ∈ E x ⊕ E x : k v k ≤ a k v k} , x ∈ L which is(a) forward invariant: D f x ( C a ( x )) ⊂ C l a ( f ( x )) and(b) expanded by forward iteration: k D f x ( v ) k ≥ l − k v k for every v ∈ C a ( x ) and there exists a cone field C b ( x ) around E which is backward invariant and expandedby backward iteration. YPERBOLIC DYNAMICAL SYSTEMS 9
Robustness.
An easy, yet very important consequence is that hyperbolic sets are robustunder small modifications of the dynamics. Indeed, suppose L is a hyperbolic set for f : M → M , and let C a ( x ) and C b ( x ) be invariant cone fields as above. The (non-invariant)decomposition E ⊕ E extends continuously to some small neighborhood U of L , andthen so do the cone fields. By continuity, conditions (a) and (b) above remain valid on U ,possibly for a slightly larger constant l . Most important, they also remain valid when f isreplaced by any other diffeomorphism g which is sufficiently C -close to it. Thus, usingthe cone field criterion once more, every compact set K ⊂ U which is invariant under g isa hyperbolic set for g . Stable manifold theorem.
Let L be a hyperbolic set for a diffeomorphism f : M → M .Assume f is of class C k . Then there exist e > < l < < e ≤ e and x ∈ L , the local stable manifold of size e W s e ( x ) = { y ∈ M : dist ( f n ( y ) , f n ( x )) ≤ e for all n ≥ } and the local unstable manifold of size e W u e ( x ) = { y ∈ M : dist ( f − n ( y ) , f − n ( x )) ≤ e for all n ≥ } are C k embedded disks, tangent at x to E sx and E ux , respectively, and satisfying • f ( W s e ( x )) ⊂ W s e ( f ( x )) and f − ( W u e ( x )) ⊂ W u e ( f − ( x )) ; • dist ( f ( x ) , f ( y )) ≤ l dist ( x , y ) for all y ∈ W s e ( x ) • dist ( f − ( x ) , f − ( y )) ≤ l dist ( x , y ) for all y ∈ W u e ( x ) • W s e ( x ) and W u e ( x ) vary continuously with the point x , in the C k topology.Then, the global stable and unstable manifolds of x , W s ( x ) = [ n ≥ f − n (cid:0) W s e ( f n ( x )) (cid:1) and W u ( x ) = [ n ≥ f n (cid:0) W u e ( f − n ( x )) (cid:1) , are smoothly immersed submanifolds of M , and they are characterized by W s ( x ) = { y ∈ M : dist ( f n ( y ) , f n ( x )) → n → ¥ } W u ( x ) = { y ∈ M : dist ( f − n ( y ) , f − n ( x )) → n → ¥ } . Shadowing property.
This crucial property of hyperbolic sets means that possible small“errors” in the iteration of the map close to the set are, in some sense, unimportant: to theresulting “wrong” trajectory, there corresponds a nearby genuine orbit of the map. Let usgive the formal statement. Recall that a hyperbolic set is compact, by definition.Given d >
0, a d - pseudo-orbit of f : M → M is a sequence { x n } n ∈ Z such thatdist ( x n + , f ( x n )) ≤ d for all n ∈ Z .Given e >
0, one says that a pseudo-orbit is e - shadowed by the orbit of a point z ∈ M ifdist ( f n ( z ) , x n ) ≤ e for all n ∈ Z . The shadowing lemma says that for any e > d > U of the hyperbolic set L such that every d -pseudo-orbit in U is e -shadowed by some orbit in U . Assuming e is sufficiently small, the shadowing orbit isactually unique. Local product structure.
In general, these shadowing orbits need not be inside th hyper-bolic set L . However, that is indeed the case if L is a maximal invariant set , that is, ifit admits some neighborhood U such that L coincides with the set of points whose orbitsnever leave U : L = \ n ∈ Z f − n ( U ) . A hyperbolic set is a maximal invariant set if and only if it has the local product structureproperty stated in the next paragraph.Let L be a hyperbolic set and e be small. If x and y are nearby points in L then the localstable manifold of x intersects the local unstable manifold of y at a unique point, denoted [ x , y ] , and this intersection is transverse. This is because the local stable manifold and thelocal unstable manifold of every point are transverse, and these local invariant manifoldsvary continuously with the point. We say that L has local product structure if there exists d > [ x , y ] belongs to L for every x , y ∈ L with dist ( x , y ) < d . Stability.
The shadowing property may also be used to prove that hyperbolic sets arestable under small perturbations of the dynamics: if L is a hyperbolic set for f then forany C -close diffeomorphism g there exists a hyperbolic set L g close to L and carrying thesame dynamical behavior.The key observation is that every orbit f n ( x ) of f inside L is a d -pseudo-orbits for g ina neighborhood U , where d is small if g is close to f and, hence, it is shadowed by someorbit g n ( z ) of g . The correspondence h ( x ) = z thus defined is injective and continuous.For any diffeomorphism g close enough to f , the orbits of x in the maximal g -invariantset L g ( U ) inside U are pseudo-orbits for f . Therefore the shadowing property above en-ables one to bijectively associate g -orbits of L g ( U ) to f -orbits in L . This provides a home-omorphism h : L g ( U ) → L which conjugates g and f on the respective hyperbolic sets: f ◦ h = h ◦ g . Thus hyperbolic maximal sets are structurally stable : the persistent dynam-ics in a neighborhood of these sets is the same for all nearby maps.If L is a hyperbolic maximal invariant set for f then its hyperbolic continuation for anynearby diffeomorphism g is also a maximal invariant set for g . Symbolic dynamics.
The dynamics of hyperbolic sets can be described through a sym-bolic coding obtained from a convenient discretization of the phase space. In a few words,one partitions the set into a finite number of subsets and assigns to a generic point in thehyperbolic set its itinerary with respect to this partition. Dynamical properties can thenbe read out from a shift map in the space of (admissible) itineraries. The precise notioninvolved is that of Markov partition.A set R ⊂ L is a rectangle if [ x , y ] ∈ R for each x , y ∈ R . A rectangle is proper if it isthe closure of its interior relative to L . A Markov partition of a hyperbolic set L is a cover R = { R , . . . , R m } of L by proper rectangles with pairwise disjoint interiors, relative to L ,and such W u ( f ( x )) ∩ R j ⊂ f ( W u ( x ) ∩ R i ) and f ( W s ( x ) ∩ R i ) ⊂ W s ( f ( x )) ∩ R j for every x ∈ int L ( R i ) with f ( x ) ∈ int L ( R j ) . The key fact is that any maximal hyperbolicset L admits Markov partitions with arbitrarily small diameter .Given a Markov partition R with sufficiently small diameter, and a sequence j = ( j n ) n ∈ Z in { , . . . , m } , there exists at most one point x = h ( j ) such that f n ( x ) ∈ R j n for each n ∈ Z . YPERBOLIC DYNAMICAL SYSTEMS 11
We say that j is admissible if such a point x does exist and, in this case, we say x admits j asan itinerary. It is clear that f ◦ h = h ◦ s , where s is the shift (left-translation) in the spaceof admissible itineraries. The map h is continuous and surjective, and it is injective on theresidual set of points whose orbits never hit the boundaries (relative to L ) of the Markovrectangles. 6. U NIFORMLY HYPERBOLIC SYSTEMS
A diffeomorphism f : M → M is uniformly hyperbolic , or satisfies the Axiom A , if thenon-wandering set W ( f ) is a hyperbolic set for f and the set Per ( f ) of periodic points isdense in W ( f ) . There is an analogous definition for smooth flows f t : M → M , t ∈ R . Thereader can find the technical details in e.g. [6], [8] and [10]. Dynamical decomposition.
The so-called “spectral” decomposition theorem of Smale al-lows for the global dynamics of a hyperbolic diffeomorphism to be decomposed into ele-mentary building blocks. It asserts that the non-wandering set splits into a finite numberof pairwise disjoint basic pieces that are compact, invariant, and dynamically indecompos-able. More precisely, the non-wandering set W ( f ) of a uniformly hyperbolic diffeomor-phism f is a finite pairwise disjoint union W ( f ) = L ∪ · · · ∪ L N of f -invariant, transitive sets L i , that are compact and maximal invariant sets. Moreover,the a -limit set of every orbit is contained in some L i and so is the w -limit set. Geodesic flows on surfaces with negative curvature.
Historically, the first importantexample of uniform hyperbolicity was the geodesic flow G t on Riemannian manifolds ofnegative curvature M . This is defined as follows.Let M be a compact Riemannian manifold. Given any tangent vector v , let g v : R → T M be the geodesic with initial condition v = g v ( ) . We denote by ˙ g v ( t ) the velocity vector attime t . Since k ˙ g v ( t ) k = k v k for all t , it is no restriction to consider only unit vectors. Thereis an important volume form on the unit tangent bundle, given by the product of the volumeelement on the manifold by the volume element induced on each fiber by the Riemannianmetric. By integration of this form, one obtains the Liouville mesure on the unit tangentbundle, which is a finite measure if the manifold itself has finite volume (including thecompact case). The geodesic flow is the flow G t : T M → T M on the unit tangent bundle T M of the manifold, defined by G t ( v ) = ˙ g v ( t ) . An important feature is that this flow leaves invariant the Liouville measure. By Poincar´erecurrence, this implies that W ( G ) = T M .A major classical result in Dynamics, due to Anosov, states that if M has negativesectional curvature then this measure is ergodic for the flow . That is, any invariant set haszero or full Liouville measure. The special case when M is a surface, had been dealt beforeby Hedlund and Hopf.The key ingredient to this theorem is to prove that the geodesic flow is uniformly hyper-bolic, in the sense we have just described, when the sectional curvature is negative. In thesurface case, the stable and unstable invariant subbundles are differentiable, which is nolonger the case in general in higher dimensions. This formidable obstacle was overcome byAnosov through showing that the corresponding invariant foliations retain, nevertheless, aweaker form of regularity property, that suffices for the proof. Let us explain this. Absolute continuity of foliations.
The invariant spaces E sx and E ux of a hyperbolic systemdepend continuously, and even H¨older continuously, on the base point x . However, in gen-eral this dependence is not differentiable, and this fact is at the origin of several importantdifficulties. Related to this, the families of stable and unstable manifolds are, usually, notdifferentiable foliations: although the leaves themselves are as smooth as the dynamicalsystem itself, the holonomy maps often fail to be differentiable. By holonomy maps wemean the projections along the leaves between two given cross-sections to the foliation.However, Anosov and Sinai observed that if the system is at least twice differentiablethen these foliations are absolutely continuous : their holonomy maps send zero Lebesguemeasure sets of one cross-section to zero Lebesgue measure sets of the other cross-section.This property is crucial for proving that any smooth measure which is invariant under atwice differentiable hyperbolic system is ergodic. For dynamical systems that are onlyonce differentiable the invariant foliations may fail to be absolutely continuous. Ergodicitystill is an open problem. Structural stability.
A dynamical system is structurally stable if it is equivalent to anyother system in a C neighborhood, meaning that there exists a global homeomorphismsending orbits of one to orbits of the other and preserving the direction of time. Moregenerally, replacing C by C r neighborhoods, any r ≥
1, one obtains the notion of C r structural stability. Notice that, in principle, this property gets weaker as r increases.The Stability Conjecture of Palis-Smale proposed a complete geometric characteriza-tion of this notion: for any r ≥ C r structurally stable systems should coincide with thehyperbolic systems having the property of strong transversality , that is, such that the stableand unstable manifolds of any points in the non-wandering set are transversal. In particu-lar, this would imply that the property of C r structural stability does not really depend onthe value of r .That hyperbolicity and strong transversality suffice for structural stability was provedin the 1970’s by Robbin, de Melo, Robinson. It is comparatively easy to prove that strongtransversality is also necessary. Thus, the heart of the conjecture is to prove that structurallystable systems must be hyperbolic. This was achieved by Ma˜n´e in the 1980’s, for C diffeomorphisms, and extended about ten years later by Hayashi for C flows. Thus a C diffeomorphism, or flow, on a compact manifold is structurally stable if and only if it isuniformly hyperbolic and satisfies the strong transversality condition. W -stability. A weaker property, called W -stability is defined requiring equivalence onlyrestricted to the non-wandering set. The W -Stability Conjecture of Palis-Smale claimsthat, for any r ≥ W -stable systems should coincide with the hyperbolic systems with nocycles , that is, such that no basic pieces in the spectral decomposition are cyclically relatedby intersections of the corresponding stable and unstable sets.The W -stability theorem of Smale states that these properties are sufficient for C r W -stability. Palis showed that the no-cycles condition is also necessary. Much later, based onMa˜n´e’s aforementioned result, he also proved that for C diffeomorphisms hyperbolicity isnecessary for W -stability. This was extended to C flows by Hayashi in the 1990’s.7. A TTRACTORS AND PHYSICAL MEASURES
A hyperbolic basic piece L i is a hyperbolic attractor if the stable set W s ( L i ) = { x ∈ M : w ( x ) ⊂ L i } YPERBOLIC DYNAMICAL SYSTEMS 13 contains a neighborhood of L i . In this case we call W s ( L i ) the basin of the attractor L i ,and denote it B ( L i ) . When the uniformly hyperbolic system is of class C , a basic piece isan attractor if and only if its stable set has positive Lebesgue measure. Thus, the union ofthe basins of all attractors is a full Lebesgue measure subset of M . This remains true for aresidual (dense G d ) subset of C uniformly hyperbolic diffeomorphisms and flows.The following fundamental result, due to Sinai, Ruelle, Bowen shows that, no matterhow complicated it may be, the behavior of typical orbits in the basin of a hyperbolicattractor is well-defined at the statistical level: any hyperbolic attractor L of a C diffeo-morphism (or flow) supports a unique invariant probability measure µ such that lim n → ¥ n n − (cid:229) j = j ( f j ( z )) = Z j dµ (2) for every continuous function j and Lebesgue almost every point x ∈ B ( L ) . The standardreference here is [3].Property (2) also means that the Sinai-Ruelle-Bowen measure µ may be “observed”: theweights of subsets may be found with any degree of precision, as the sojourn-time of anyorbit picked “at random” in the basin of attraction: µ ( V ) = fraction of time the orbit of z spends in V for typical subsets V of M (the boundary of V should have zero µ -measure), and forLebesgue almost any point z ∈ B ( L ) . For this reason µ is called a physical measure .It also follows from the construction of these physical measures on hyperbolic attrac-tors that they depend continuously on the diffeomorphism (or the flow). This statisticalstability is another sense in which the asymptotic behavior is stable under perturbations ofthe system, distinct from structural stability.There is another sense in which this measure is “physical” and that is that µ is the zero-noise limit of the stationary measures associated to the stochastic processes obtained byadding small random noise to the system. The idea is to replace genuine trajectories by“random orbits” ( z n ) n , where each z n + is chosen e -close to f ( z n ) . We speak of stochasticstability if, for any continuous function j , the random time averagelim n → ¥ n n − (cid:229) j = j ( z j ) is close to R j dµ for almost all choices of the random orbit.One way to construct such random orbits is through randomly perturbed iterations, asfollows. Consider a family of probability measures n e in the space of diffeomorphisms,such that each n e is supported in the e -neighborhood of f . Then, for each initial state z define z n + = f n + ( z n ) , where the diffeomorphisms f n are independent random variableswith distribution law n e . A probability measure h e on the basin B ( L ) is stationary if itsatisfies h e ( E ) = Z h e ( g − ( E )) d n e ( g ) . Stationary measures always exist, and they are often unique for each small e >
0. Thenstochastic stability corresponds to having h e converging weakly to µ when the noise level e goes to zero.The notion of stochastic stability goes back to Kolmogorov and Sinai. The first results,showing that uniformly hyperbolic systems are stochastically stable, on the basin of eachattractor, were proved in the 1980’s by Kifer and Young. Let us point out that physical measures need not exist for general systems. A simplecounter-example, attributed to Bowen, is described in Figure 3: time averages diverge overany of the spiraling orbits in the region bounded by the saddle connections. Notice that thesaddle connections are easily broken by arbitrarily small perturbations of the flow. Indeed,no robust examples are known of systems whose time-averages diverge on positive volumesets.
A Bz F IGURE
3. A planar flow with divergent time averages8. O
BSTRUCTIONS TO HYPERBOLICITY
Although uniform hyperbolicity was originally intended to encompass a residual or, atleast, dense subset of all dynamical systems, it was soon realized that this is not the case:many important examples fall outside its realm. There are two main mechanisms that yieldrobustly non-hyperbolic behavior, that is, whole open sets of non-hyperbolic systems.
Heterodimensional cycles.
Historically, the first such mechanism was the coexistence ofperiodic points with different Morse indices (dimensions of the unstable manifolds) in-side the same transitive set. See Figure 4. This is how the first examples of C -opensubsets of non-hyperbolic diffeomorphisms were obtained by Abraham, Smale on mani-folds of dimension d ≥
3. It was also the key in the constructions by Shub and Ma˜n´e ofnon-hyperbolic, yet robustly transitive diffeomorphisms, that is, such that every diffeomor-phism in a C neighborhood has dense orbits. q p p F IGURE
4. A heterodimensional cycleFor flows, this mechanism may assume a novel form, because of the interplay betweenregular orbits and singularities (equilibrium points). That is, robust non-hyperbolicity maystem from the coexistence of regular and singular orbits in the same transitive set. The first,and very striking example was the geometric Lorenz attractor proposed by Afraimovich,Bykov, Shil’nikov and Guckenheimer, Williams to model the behavior of the Lorenz equa-tions, that we shall discuss later.
YPERBOLIC DYNAMICAL SYSTEMS 15
Homoclinic tangencies.
Of course, heterodimensional cycles may exist only in dimension3 or higher. The first robust examples of non-hyperbolic diffeomorphisms on surfaces wereconstructed by Newhouse, exploiting the second of these two mechanisms: homoclinictangencies, or non-transverse intersections between the stable and the unstable manifold ofthe same periodic point. See Figure 5. q q
Hpp F IGURE
5. Homoclinic tangenciesIt is important to observe that individual homoclinic tangencies are easily destroyedby small perturbations of the invariant manifolds. To construct open examples of surfacediffeomorphisms with some tangency, Newhouse started from systems where the tangencyis associated to a periodic point inside an invariant hyperbolic set with rich geometricstructure. This is illustrated on the right hand side of Figure 5. His argument requires avery delicate control of distortion, as well as of the dependence of the fractal dimensionon the dynamics. Actually, for this reason, his construction is restricted to the C r topologyfor r ≥
2. A very striking consequence of this construction is that these open sets exhibit coexistence of infinitely many periodic attractors , for each diffeomorphism on a residualsubset. A detailed presentation of his result and consequences is given in [9].Newhouse’s conclusions have been extended in two ways. First, by Palis, Viana, fordiffeomorphisms in any dimension, still in the C r topology with r ≥
2. Then, by Bonatti,D´ıaz, for C diffeomorphisms in any dimension larger or equal than 3. The case of C diffeomorphisms on surfaces remains open. As a matter of fact, in this setting it is stillunknown whether uniform hyperbolicity is dense in the space of all diffeomorphisms.9. P ARTIAL HYPERBOLICITY
Several extensions of the theory of uniform hyperbolicity have been proposed, allow-ing for more flexibility, while keeping the core idea: splitting of the tangent bundle intoinvariant subbundles. We are going to discuss more closely two such extensions.On the one hand, one may allow for one or more invariant subbundles along whichthe derivative exhibits mixed contracting/neutral/expanding behavior. This is genericallyreferred to as partial hyperbolicity , and a standard reference is the book [5]. On the otherhand, while requiring all invariant subbundles to be either expanding or contraction, onemay relax the requirement of uniform rates of expansion and contraction. This is usuallycalled non-uniform hyperbolicity . A detailed presentation of the fundamental results aboutthis notion is available e.g. in [6]. In this section we discuss the first type of condition. Thesecond one will be dealt with later.
Dominated splittings.
Let f : M → M be a diffeomorphism on a closed manifold M and K be any f -invariant set. A continuous splitting T x M = E ( x ) ⊕ · · · ⊕ E k ( x ) , x ∈ K of thetangent bundle over K is dominated if it is invariant under the derivative D f and thereexists ℓ ∈ N such that for every i < j , every x ∈ K , and every pair of unit vectors u ∈ E i ( x ) and v ∈ E j ( x ) , one has k D f ℓ x · u kk D f ℓ x · v k <
12 (3)and the dimension of E i ( x ) is independent of x ∈ K for every i ∈ { , . . . , k } . This definitionmay be formulated, equivalently, as follows: there exist C > l < u ∈ E i ( x ) and v ∈ E j ( x ) , one has k D f nx · u kk D f nx · v k < C l n for all n ≥ . Let f be a diffeomorphism and K be an f -invariant set having a dominated splitting T K M = E ⊕ · · · ⊕ E k . We say that the splitting and the set K are • partially hyperbolic the derivative either contracts uniformly E or expands uni-formly E k : there exists ℓ ∈ N such thateither k D f ℓ | E k <
12 or k ( D f ℓ | E k ) − k < . • volume hyperbolic if the derivative either contracts volume uniformly along E orexpands volume uniformly along E k : there exists ℓ ∈ N such thateither | det ( D f ℓ | E ) | <
12 or | det ( D f ℓ | E k ) | > . The diffeomorphism f is partially hyperbolic/volume hyperbolic if the ambient space M is a partially hyperbolic/volume hyperbolic set for f . Invariant foliations.
An crucial geometric feature of partially hyperbolic systems is theexistence of invariant foliations tangent to uniformly expanding or uniformly contractinginvariant subbundles: assuming the derivative contracts E uniformly, there exists a uniquefamily F s = { F s ( x ) : x ∈ K } of injectively C r immersed submanifolds tangent to E at ev-ery point of K, satisfying f ( F s ( x )) = F s ( f ( x )) for all x ∈ K, and which are uniformlycontracted by forward iterates of f .
This is called strong-stable foliation of the diffeomor-phism on K . Strong-unstable foliations are defined in the same way, tangent to the invariantsubbundle E k , when it is uniformly expanding.As in the purely hyperbolic setting, a crucial ingredient in the ergodic theory of par-tially hyperbolic systems is the fact that strong-stable and strong-unstable foliations areabsolutely continuous, if the system is at least twice differentiable. Robustness and partial hyperbolicity.
Partially hyperbolic systems have been studiedsince the 1970’s, most notably by Brin, Pesin and Hirsch, Pugh, Shub. Over the last decadethey attracted much attention as the key to characterizing robustness of the dynamics. Moreprecisely, let L be a maximal invariant set of some diffeomorphism f : L = \ n ∈ Z f n ( U ) for some neighborhood U of L .The set L is robust , or robustly transitive , if its continuation L g = ∩ n ∈ Z g n ( U ) is transitivefor all g in a neighborhood of f . There is a corresponding notion for flows.As we have already seen, hyperbolic basic pieces are robust. In the 1970’s, Ma˜n´e ob-served that the converse is also true when M is a surface, but not anymore if the dimension YPERBOLIC DYNAMICAL SYSTEMS 17 of M is at least 3. Counter-examples in dimension 4 had been given before by Shub. Aseries of results of Bonatti, D´ıaz, Pujals, Ures in the 1990’s clarified the situation in alldimensions: robust sets always admit some dominated splitting which is volume hyper-bolic; in general, this splitting needs not be partially hyperbolic, except when the ambientmanifold has dimension 3. Lorenz-like strange attractors.
Parallel results hold for flows on 3-dimensional mani-folds. The main motivation are the so-called Lorenz-like strange attractors, inspired by thefamous differential equations ˙ x = − s x + s y s = y = r x − y − xz r = z = xy − b z b = / any robustinvariant set of a flow in dimension is singular hyperbolic . Moreover, if the robust setcontains equilibrium points then it must be either an attractor or a repeller . A detailedpresentation of this and related results is given in [1].An invariant set L of a flow in dimension 3 is singular hyperbolic if it is a partiallyhyperbolic set with splitting E ⊕ E such that the derivative is volume contracting along E and volume expanding along E . Notice that one of the subbundles E or E mustbe one-dimensional, and then the derivative is, actually, either norm contracting or normexpanding along this subbundle. Singular hyperbolic sets without equilibria are uniformlyhyperbolic: the 2-dimensional invariant subbundle splits as the sum of the flow directionwith a uniformly expanding or contracting one-dimensional invariant subbundle.10. N ON - UNIFORM HYPERBOLICITY - L
INEAR THEORY
In its linear form, the theory of non-uniform hyperbolicity goes back to Lyapunov, andis founded on the multiplicative ergodic theorem of Oseledets. Let us introduce the mainideas, whose thorough development can be found in e.g. [4], [6] and [7].The
Lyapunov exponents of a sequence { A n , n ≥ } of square matrices of dimension d ≥
1, are the values of l ( v ) = lim sup n → ¥ n log k A n · v k (5)over all non-zero vectors v ∈ R d . For completeness, set l ( ) = − ¥ . It is easy to see that l ( cv ) = l ( v ) and l ( v + v ′ ) ≤ max { l ( v ) , l ( v ′ ) } for any non-zero scalar c and any vectors v , v ′ . It follows that, given any constant a , the set of vectors satisfying l ( v ) ≤ a is a vectorsubspace. Consequently, there are at most d Lyapunov exponents, henceforth denoted by l < · · · < l k − < l k , and there exists a filtration F ⊂ F ⊂ · · · ⊂ F k − ⊂ F k = R d intovector subspaces, such that l ( v ) = l i for all v ∈ F i \ F i − and every i = , . . . , k (write F = { } ). In particular, the largest exponent is given by l k = lim sup n → ¥ n log k A n k . (6)One calls dim F i − dim F i − the multiplicity of each Lyapunov exponent l i .There are corresponding notions for continuous families of matrices A t , t ∈ ( , ¥ ) , tak-ing the limit as t goes to infinity in the relations (5) and (6). Lyapunov stability.
Consider the linear differential equation˙ v ( t ) = B ( t ) · v ( t ) (7)where B ( t ) is a bounded function with values in the space of d × d matrices, defined for all t ∈ R . The theory of differential equations ensures that there exists a fundamental matrixA t , t ∈ R such that v ( t ) = A t · v is the unique solution of (7) with initial condition v ( ) = v .If the Lyapunov exponents of the family A t , t > v ( t ) ≡ w ( t ) = B ( t ) · w + F ( t , w ) with k F ( t , w ) k ≤ const k w k + c , c >
0. That is, the trivial solution w ( t ) ≡ v by elements v ∧ · · · ∧ v l of any l th exterior power of R d , 1 ≤ l ≤ d .By definition, the norm of an l -vector v ∧ · · · ∧ v l is the volume of the parallelepipeddetermined by the vectors v , . . . , v k . This condition is usually tricky to check in specificsituations. However, the multiplicative ergodic theorem of V. I. Oseledets asserts that,for very general matrix-valued stationary random processes, regularity is an almost sureproperty. Multiplicative ergodic theorem.
Let f : M → M be a measurable transformation, pre-serving some measure µ , and let A : M → GL ( d , R ) be any measurable function such thatlog k A ( x ) k is µ -integrable. The Oseledets theorem states that Lyapunov exponents existfor the sequence A n ( x ) = A ( f n − ( x )) · · · A ( f ( x )) A ( x ) for µ -almost every x ∈ M . More pre-cisely, for µ -almost every x ∈ M there exists k = k ( x ) ∈ { , . . . , d } , a filtration F x ⊂ F x ⊂ · · · ⊂ F k − x ⊂ F kx = R d , and numbers l ( x ) < · · · < l k ( x ) such thatlim n → ¥ n log k A n ( x ) · v k = l i ( x ) for all v ∈ F ix \ F i − x and i ∈ { , . . ., k } . More generally, this conclusion holds for any vectorbundle automorphism V → V over the transformation f , with A x : V x → V f ( x ) denotingthe action of the automorphism on the fiber of x . YPERBOLIC DYNAMICAL SYSTEMS 19
The Lyapunov exponents l i ( x ) , and their number k ( x ) , are measurable functions of x and they are constant on orbits of the transformation f . In particular, if the measure µ isergodic then k and the l i are constant on a full µ -measure set of points. The subspaces F ix also depend measurably on the point x and are invariant under the automorphism: A ( x ) · F ix = F if ( x ) . It is in the nature of things that, usually, these objects are not defined everywhere and theydepend discontinuously on the base point x .When the transformation f is invertible one obtains a stronger conclusion, by applyingthe previous result also to the inverse automorphism: assuming that log k A ( x ) − k is also in L ( µ ) , one gets that there exists a decomposition V x = E x ⊕ · · · ⊕ E kx , defined at almost every point and such that A ( x ) · E ix = E if ( x ) andlim n →± ¥ n log k A n ( x ) · v k = l i ( x ) for all v ∈ E ix different from zero and all i ∈ { , . . . , k } . These Oseledets subspaces E ix arerelated to the subspaces F ix through F jx = ⊕ ji = E ix . Hence, dim E ix = dim F ix − dim F i − x is the multiplicity of the Lyapunov exponent l i ( x ) .The angles between any two Oseledets subspaces decay sub-exponentially along orbitsof f : lim n →± ¥ n log angle ( M i ∈ I E if n ( x ) , M j / ∈ I E jf n ( x ) ) = I ⊂ { , . . . , k } and almost every point. These facts imply the regularity conditionmentioned previously and, in particular,lim n →± ¥ n log | det A n ( x ) | = k (cid:229) i = l i ( x ) dim E ix Consequently, if det A ( x ) = ON - UNIFORMLY HYPERBOLIC SYSTEMS
The Oseledets theorem applies, in particular, when f : M → M is a C diffeomorphismon some compact manifold and A ( x ) = D f x . Notice that the integrability conditions areautomatically satisfied, for any f -invariant probability measure µ , since the derivative of f and its inverse are bounded in norm.Lyapunov exponents yield deep geometric information on the dynamics of the diffeo-morphism, especially when they do not vanish. We call µ a hyperbolic measure if allLyapunov exponents are non-zero at µ -almost every point. By non-uniformly hyperbolicsystem we shall mean a diffeomorphism f : M → M together with some invariant hyper-bolic measure.A theory initiated by Pesin provides fundamental geometric information on this classof systems, especially existence of stable and unstable manifolds at almost every point which form absolutely continuous invariant laminations. For most results, one needs thederivative D f to be H¨older continuous: there exists c > k D f x − D f y k ≤ const · d ( x , y ) c . These notions extend to the context of flows essentially without change, except that onedisregards the invariant line bundle given by the flow direction (whose Lyapunov exponentis always zero). A detailed presentation can be found in e.g. [6].
Stable manifolds.
An essential tool is the existence of invariant families of local stablesets and local unstable sets, defined at µ -almost every point. Assume µ is a hyperbolicmeasure. Let E ux and E sx be the sums of all Oseledets subspaces corresponding to positive,respectively negative, Lyapunov exponents, and let t x > x .Pesin’s stable manifold theorem states that, for µ-almost every x ∈ M, there exists a C embedded disk W sloc ( x ) tangent to E sx at x and there exists C x > such that dist ( f n ( y ) , f n ( x )) ≤ C x e − n t x · dist ( y , x ) for all y ∈ W sloc ( x ) . Moreover, the family { W sloc ( x ) } is invariant, in the sense that f ( W sloc ( x )) ⊂ W sloc ( f ( x )) for µ -almost every x . Thus, one may define global stable manifolds W s ( x ) = ¥ [ n = f − n (cid:0) W sloc ( x ) (cid:1) for µ -almost every x .In general, the local stable disks W s ( x ) depend only measurably on x . Another key differ-ence with respect to the uniformly hyperbolic setting is that the numbers C x and t x can notbe taken independent of the point, in general. Likewise, one defines local and global un-stable manifolds, tangent to E ux at almost every point. Most important for the applications,both foliations, stable and unstable, are absolutely continuous.In the remaining sections we briefly present three major results in the theory of non-uniform hyperbolicity: the entropy formula, abundance of periodic orbits, and exact di-mensionality of hyperbolic measures. The entropy formula.
The entropy of a partition P of M is defined by h µ ( f , P ) = lim n → ¥ n H µ ( P n ) , where P n is the partition into sets of the form P = P ∩ f − ( P ) ∩ · · · ∩ f − n ( P n ) with P j ∈ P and H µ ( P n ) = (cid:229) P ∈ P n − µ ( P ) log µ ( P ) . The
Kolmogorov-Sinai entropy h µ ( f ) of the system is the supremum of h µ ( f , P ) over allpartitions P with finite entropy. The Ruelle-Margulis inequality says that h µ ( f ) is boundedabove by the averaged sum of the positive Lyapunov exponents. A major result of thetheorem, Pesin’s entropy formula, asserts that if the invariant measure µ is smooth (forinstance, a volume element) then the entropy actually coincides with the averaged sum ofthe positive Lyapunov exponents h µ ( f ) = Z (cid:0) k (cid:229) j = max { , l j } (cid:1) dµ . A complete characterization of the invariant measures for which the entropy formula istrue was given by F. Ledrappier and L. S. Young.
YPERBOLIC DYNAMICAL SYSTEMS 21
Periodic orbits and entropy.
It was proved by A. Katok that periodic motions are alwaysdense in the support of any hyperbolic measure. More than that, assuming the measure isnon-atomic, there exist Smale horseshoes H n with topological entropy arbitrarily close tothe entropy h µ ( f ) of the system. In this context, the topological entropy h ( f , H n ) may bedefined as the exponential rate of growthlim k → ¥ k log { x ∈ H n : f k ( x ) = x } . of the number of periodic points on H n . Dimension of hyperbolic measures.
Another remarkable feature of hyperbolic measuresis that they are exact dimensional : the pointwise dimension d ( x ) = lim r → log µ ( B r ( x )) log r exists at almost every point, where B r ( x ) is the neighborhood of radius r around x . Thisfact was proved by L. Barreira, Ya. Pesin, and J. Schmeling. Note that this means that themeasure µ ( B r ( x )) of neighborhoods scales as r d ( x ) when the radius r is small.12. F UTURE DIRECTIONS
The theory of uniform hyperbolicity showed that dynamical systems with very complexbehavior may be amenable to a very precise description of their evolution, especially inprobabilistic terms. It was most successful in characterizing structural stability, and alsoestablished a paradigm of how general ”chaotic” systems might be approached. A vastresearch program has been going on in the last couple of decades or so, to try and buildsuch a global theory of complex dynamical evolution, where notions such as partial andnon-uniform hyperbolicity play a central part. The reader is referred to the bibliography,especially the book [2] for a review of much recent progress.R
EFERENCES[1] V. Araujo and M. J. Pacifico.
Three Dimensional Flows . XXV Brazillian Mathematical Colloquium. IMPA,Rio de Janeiro, 2007.[2] C. Bonatti, L. J. D´ıaz, and M. Viana.
Dynamics beyond uniform hyperbolicity , volume 102 of
Encyclopaediaof Mathematical Sciences . Springer-Verlag, 2005.[3] R. Bowen.
Equilibrium states and the ergodic theory of Anosov diffeomorphisms , volume 470 of
Lect. Notesin Math.
Springer Verlag, 1975.[4] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sina˘ı.
Ergodic theory , volume 245 of
Grundlehren der Mathematis-chen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, 1982.[5] M. Hirsch, C. Pugh, and M. Shub.
Invariant manifolds , volume 583 of
Lect. Notes in Math.
Springer Verlag,1977.[6] A. Katok and B. Hasselblatt.
Introduction to the modern theory of dynamical systems . Cambridge UniversityPress, 1995.[7] R. Ma˜n´e.
Ergodic theory and differentiable dynamics . Springer Verlag, 1987.[8] J. Palis and W. de Melo.
Geometric theory of dynamical systems. An introduction . Springer Verlag, 1982.[9] J. Palis and F. Takens.
Hyperbolicity and sensitive-chaotic dynamics at homoclinic bifurcations . CambridgeUniversity Press, 1993.[10] M. Shub.
Global stability of dynamical systems . Springer Verlag, 1987.CMUP, R
UA DO C AMPO A LEGRE
ORTO , P
ORTUGAL
E-mail address : [email protected] IM-UFRJ, C.P. 68.530, CEP 21.945-970, R
IO DE J ANEIRO , B
RAZIL
E-mail address : [email protected] IMPA, E
STRADA D ONA C ASTORINA
IO DE J ANEIRO , B
RASIL
E-mail address : [email protected]@impa.br