Flexibility of Lyapunov exponents

We outline the flexibility program in smooth dynamics, focusing on flexibility of Lyapunov exponents for volume-preserving diffeomorphisms. We prove flexibility results for Anosov diffeomorphisms admitting dominated splittings into one-dimensional bundles.


Introduction
1.1. The flexibility program. Important attributes of smooth dynamical systems such as entropies and Lyapunov characteristic exponents with respect to a relevant invariant measure (i.e. a volume, an SRB measure, or a measure of maximal entropy) reflect asymptotic behavior of orbits and with rare exceptions cannot be calculated in a closed form. Exceptions are systems of algebraic origin, such as translations on homogeneous spaces and affine maps on compact abelian groups and, in the case of topological entropy, structurally stable discrete time hyperbolic systems where topological entropy can be calculated using an algebraic or symbolic model. Beyond that there are few general relations for various classes of systems, in the form of equalities or inequalities, either involving only dynamical characteristics themselves or relating those with other quantities coming from geometry, topology or analysis. Let us list some of those relations. Those marked with an asterisk are valid for topological dynamical systems on compact spaces; others require some smoothness assumptions. We refer to original sources only if no standard monograph or textbook exposition is available.
‚ ‚ For Anosov systems on infranilmanifolds Shub entropy inequality becomes equality (for the torus case, see [27,Theorem 18.6.1]). ‚ Conformal inequality for entropies for geodesic flows on manifolds of negative curvature [26]. At a more basic level, preservation of a geometric structure imposes restrictions on dynamical invariants. For example, for a volume-preserving system the sum of Lyapunov characteristic exponents is zero, for a holomorphic system all exponents have even multiplicity, and for a symplectic map exponents come in pairs˘λ.
The general paradigm of flexibility can be rather vaguely formulated as follows: (F) Under properly understood general restrictions (like those listed or mentioned above), within a fixed class of smooth dynamical systems dynamical invariants take arbitrary values. In the context of smooth ergodic theory, the most natural flexibility problem concerns Lyapunov exponents for volume-preserving systems with respect to the volume measure. We restrict our discussion to the classical discrete time invertible dynamical systems, i.e. actions of Z. The continuous time case in some key situations follows directly from the discrete one via the suspension construction, in the others can be treated in a parallel way and in certain respects it is easier since the homotopy restrictions (see below) do not appear.
The case of multidimensional time is very different. There the phenomenon of rigidity that in a sense is complementary to flexibility is prevalent: see e.g. [28].
Other instances of the flexibility program are briefly discussed in Subsection 8.2. xist and hence are constant µ-almost everywhere. They are called Lyapunov characteristic exponents or often simply Lyapunov exponents of f with respect to µ and are denoted by λ 1,µ pf q ě¨¨¨ě λ d,µ pf q. For the full Oseledets Theorem (which also describes the growth of tangent vectors), see e.g. [3,7]. The Lyapunov spectrum is defined as the vector (1.2) λ µ pf q :"`λ 1,µ pf q, . . . , λ d,µ pf q˘.
We say that this spectrum is simple if none of these numbers is repeated. Let m be a smooth volume measure, normalized so that mpM q " 1. The particular choice is not important, since for any pair of such measures, there exists a diffeomorphism taking one to the other [33], [27,Theorem 5.1.27]. Given r P t1, 2, . . . , 8u, let Diff r m pM q denote the set of m-preserving (also called conservative) diffeomorphisms f : M Ñ M of class C r . We will discuss the case when f is ergodic with respect to m; for simplicity we write λ i pf q " λ i,m pf q, λpf q " λ m pf q. We always have ř d i"1 λ i pf q " 0. Now we formulate and discuss several representative questions concerning flexibility of Lyapunov exponents for general conservative diffeomorphisms. Conjecture 1.1 (Weak flexibility -general). Given any list of numbers ξ 1 ě¨¨¨ě ξ d with ř d i"1 ξ i " 0, there exists an ergodic diffeomorphism f P Diff 8 m pM q such that λpf q " pξ 1 , . . . , ξ d q. Conjecture 1.2 (Strong flexibility -general). Given a connected component C Ď Diff 8 m pM q and any list of numbers ξ 1 ě¨¨¨ě ξ d with ř d i"1 ξ i " 0, there exists an ergodic diffeomorphism f P C such that λpf q " pξ 1 , . . . , ξ d q.
If all exponents are equal to zero then Conjecture 1.1 is known; this has been proved long ago [2,1]. In this case Conjecture 1.2 holds for the identity component provided that the dimension is at least 3. 1 On an opposite direction, the existence of conservative ergodic (actually Bernoulli) smooth diffeomorphisms without zero Lyapunov exponents on any manifold was established by Dolgopyat and Pesin [18] (the 2-dimensional case was settled earlier [25]). These examples are homotopic to the identity.
In this paper we will not attack Conjectures 1.1 and 1.2 directly. Instead, we will establish flexibility results for a particular and more tractable class of systems, namely Anosov diffeomorphisms admitting simple dominated splitting. Nevertheless, we believe that our methods (combined with techniques from the aforementioned works) should provide the basis for an approach on the conjectures, at least under some restrictions.
1.3. The Anosov case. Anosov systems represent a natural class for the flexibility analysis. We work with conservative Anosov diffeomophisms which are at least C 2 ; then, by a classical theorem of Anosov and Sinai, the volume measure m is ergodic.
All known Anosov diffeomorphisms are topologically conjugate to automorphisms of infranilmanifolds that include tori and nilmanifolds as special cases. Hence the metric entropy with respect to invariant volume (equal to the sum of positive Lyapunov exponents) does not exceed the sum of positive Lyapunov exponents for the corresponding automorphism that is 1 Existence of ergodic diffeomorphisms with zero exponents on any manifold with a non-trivial action of the circle S 1 including the 2-disc D 2 , 2-sphere S 2 , the annulus, and the Klein bottle, has been established in the paper [2], which can be viewed as the earliest work on flexibility. However in the case of determined by induced automorphism of the fundamental group. The main flexibility question is whether this is the only restriction.
In order to simplify the notation we restrict our discussion to the torus case. Let L P GLpd, Zq and assume that L is hyperbolic, i.e. the absolute values of all of its eigenvalues are different from one. The matrix L determines the automorphism F L of the torus T d :" R d {Z d , which is a conservative Anosov diffeomorphism.
Every Anosov diffeomorphism f of T d (conservative of not) is homotopic and, moreover, topologically conjugate via a homeomorphism isotopic to identity, to an automorphism F L , where L is a hyperbolic matrix [27, Theorem 18.6.1]. In fact, L is the matrix of the automorphism induced by f on the fundamental group of T d , which is naturally isomorphic to Z d . 2 Given a hyperbolic matrix L P GLpd, Zq, the Lyapunov spectrum of the automorphism λpF L q is the vector λpLq whose entries λ 1 pLq ě¨¨¨ě λ d pLq are the logarithms of the absolute values of the eigenvalues of L, repeated according to multiplicity. The number u " upLq of positive elements in this list is called the unstable index of L; so λ u pLq ą 0 ą λ u`1 pLq. The quantity ř u i"1 λ i pLq is equal to both topological entropy h top pF L q and to the metric entropy h m pF L q with respect to Lebesgue measure m on T d . Therefore for any conservative Anosov using Pesin's formula, the variational principle, and the above-mentioned topological conjugacy. 3 Are there other restrictions on the spectrum of f ? We pose the following: Problem 1.3 (Strong flexibility -Anosov). Let L P GLpd, Zq be a hyperbolic matrix, and let u be its unstable index. Given any list of numbers ξ 1 ě¨¨¨ě ξ u ą 0 ą ξ u`1 ě¨¨¨ě ξ d such that does there exist a conservative Anosov diffeomorphism f homotopic (and hence topologically conjugate) to F L such that λpf q " pξ 1 , . . . , ξ d q?
Regularity of f may vary but it does not seem likely that the answer depends on regularity, at least above C 1 .
For d " 2 (and so u " 1), the problem reduces to existence of Anosov diffeomorphisms on T 2 with any positive value of metric entropy below the 2 However, for large enough d is not always true that there is an homotopy between f and FL consisting of Anosov diffeomorphisms: see [21]. 3 In reality, the inequality ř u i"1 ξi ď ř u i"1 λipLq also holds when f is only C 1 ; indeed it follows from the C 1`ε case using C 1 -continuity of the right-hand side and Avila's regularization [4]. topological entropy. Here the answer is positive. It is not difficult to produce such examples even in the real-analytic category by a fairly straightforward global twist construction. Existence of C 8 examples also follows from our Theorem 1.5 below.
In its weak version, i.e. without considerations about homotopy, the flexibility problem is likely to have a positive solution: Conjecture 1.4 (Weak flexibility -Anosov). Given any list of nonzero numbers ξ 1 ě¨¨¨ě ξ d such that ř d i"1 ξ i " 0, there exists an Anosov diffeomorphism of T d such that λpf q " pξ 1 , . . . , ξ d q.
The case of this conjecture with strict inequalities easily follows from our main result: see Corollary 1.6 below.
is called dominated if each of the bundles dominates the next. This means that given a Riemannian metric, there exists n 0 ě 1 such that for every x P M and all unit vectors v 1 P E 1 pxq, . . . , v k P E k pxq, we have It is always possible to find an "adapted" Riemannian metric for which n 0 " 1: see [23]. Dominated splittings are automatically continuous, and their existence is a C 1 -open condition; see e.g. [14, § B.1] for these and other properties. We say that a dominated splitting is simple if all the subbundles E j are one-dimensional (and so k " d). In this case the Oseledets splitting coincides with E 1 '¨¨¨' E d almost everywhere, the Lyapunov spectrum is simple, and the Lyapunov exponents with respect to invariant volume m are given by integrals: In particular, in the class of diffeomorphisms admitting simple dominated splitting, the Lyapunov exponents depend continuously on the dynamics, and therefore the flexibility analysis becomes somewhat simpler. On the other hand, in the absence of domination, small perturbations (with respect to the C 1 topology) of the dynamics may have a large effect on the Lyapunov spectrum and even send all Lyapunov exponents to zero [10,13] (but the C 2 norm of such a perturbation generally explodes [30]). Existence of a dominated splitting also imposes restrictions on the topology of the manifold.
1.5. Formulation of results. Let us recall the classical notion of majorization, which has a wide range of applications (see e.g. [31]). 4 4 See [12] for another instance where majorization plays a role in the perturbation of Lyapunov exponents.
Intuitively, ξ ě η means that the entries of η are obtained from those of ξ by a process of "mixing". Let us state this precisely: If ξ majorizes η then there exists a doubly-stochastic dˆd matrix P such that η " P ξ; conversely, given a ordered vector ξ and a doubly-stochastic matrix P , the vector obtained by reordering the entries of P ξ is majorized by ξ -see [31,Theorem B.2].
We now state the main result of this paper. Recall that M is a smooth compact connected manifold of dimension d ě 2, and m is a smooth volume measure, normalized so that mpM q " 1; note that we do not assume that M is a torus (nor even an infranilmanifold). The unstable index of an Anosov diffeomorphism is the dimension of its unstable bundle. Theorem 1.5. Let r P t2, 3, . . . , 8u, and let f P Diff r m pM q be a conservative Anosov C r -diffeomorphism with simple dominated splitting. Let ξ P R d be such that: (a) ξ 1 ą¨¨¨ą ξ u ą 0 ą ξ u`1 ą¨¨¨ą ξ d , where u is the unstable index of f ; (b) ξ ă λpf q, that is, ξ is strictly majorized by λpf q. Then there is a continuous path pf t q tPr0,1s in Diff r m pM q such that: ‚ f 0 " f ; ‚ each f t is Anosov with simple dominated splitting; ‚ λpf 1 q " ξ. Corollary 1.6 (Anosov diffeomorphisms display all hyperbolic simple Lyapunov spectra). Given any list of nonzero numbers ξ 1 ą¨¨¨ą ξ d whose sum is equal to 0, there exists a conservative Anosov C 8 diffeomorphism of T d with simple dominated splitting such that λpf q " pξ 1 , . . . , ξ d q.
Corollary 1.6 is obtained as follows: first we take an Anosov linear automorphism whose spectrum is simple and "large" with respect to the majorization partial order; then Theorem 1.5 allows us to deform the linear automorphism and obtain a conservative Anosov diffeomorphism with the desired Lyapunov spectrum. (See Subsection 7.1 for full details.) Note that as a consequence of Corollary 1.6 we obtain a positive solution of the general weak flexibility Conjecture 1.1 on tori for simple spectra: if the desired spectrum contains 0 then we just take the product f " gˆR θ of an appropriate Anosov map g on T d´1 and an irrational rotation R θ on T; this f is ergodic since g is mixing.
While condition (b) in Theorem 1.5 asks for strict majorization, there are specific situations where this requirement can be relaxed to ordinary majorization. This is demonstrated by the next theorem, which also shows that the majorization condition is indeed necessary: Theorem 1.7. Let F L be a Anosov linear automorphism of T 3 with simple Lyapunov spectrum and unstable index u. Given ξ " pξ 1 , ξ 2 , ξ 3 q P R 3 , then there exists an Anosov diffeomorphism f P Diff 8 m pT 3 q homotopic (and hence topologically conjugate) to F L and with simple dominated splitting such that λpf q " ξ if and only if (1.9) ξ 1 ą ξ 2 ą ξ 3 , ξ u ą 0 ą ξ u`1 , and ξ ď λpLq .
Furthermore, one can choose an homotopy between F L and f consisting of conservative smooth Anosov diffeomorphisms with simple dominated splitting.
In d " 3, the condition ξ ď λpLq is strictly stronger than the "entropy condition" (1.4). Therefore, if Problem 1.3 has a positive solution, it necessarily involves Anosov diffeomorphisms without a simple dominated splitting, even in the case of simple Lyapunov spectra. The existence of a single conservative Anosov diffeomorphism f : T 3 Ñ T 3 whose spectrum ξ " λpf q does not satisfy the entropy condition (1.4) is already a very interesting question.
Let us note that Hu, Jiang, and Jiang [24] have constructed deformations of conservative Anosov diffeomorphisms and of conservative expanding endomorphisms having arbitrarily small metric entropy. In the case of diffeomorphisms with dominated splittings, their result follows from Theorem 1.5. Their construction is very different from ours.
1.6. Comments on the proofs. Let us summarize the ideas of the proof of Theorem 1.5. Motivated by the work of Shub and Wilkinson [41], Baraviera and Bonatti [6] have proved the following "local flexibility" result: given a conservative stably ergodic partially hyperbolic diffeomorphism, one can perturb it so that the sum of central Lyapunov exponents becomes different from zero. Their idea was to perturb the diffeomorphism on a small ball around a non-periodic point (so to avoid fast returns) by rotating on a centerunstable plane so that the central bundle borrows some expansion from the unstable bundle. Actually, their argument allows to slightly mix Lyapunov exponents in any pair of consecutive bundles in a dominated splitting, while the Lyapunov exponents in the other bundles move extremely little. 6 So it is conceivable that with a sequence of Baraviera-Bonatti perturbations one could mix Lyapunov exponents basically at will. Though this idea is ultimately correct, several difficulties need to be overcame in order to turn it into a proof of Theorem 1.5. First, how can we ensure that the effect of the perturbations is not too weak ? Second, how can we obtain a prescribed Lyapunov spectrum exactly?
Let us discuss how to overcome the first difficulty. Instead of using a single ball as the support of a perturbation, we select several small balls whose union has a large first return time but non-negligible measure: this is done with a standard tower construction (in the style of [10,13], for instance). In each of these balls one composes with the same "model" perturbation of the identity that rotates the appropriate plane. However, this trick by itself is not sufficient to conclude. If the domination or the hyperbolicity gets weak, the rotations should be smaller and their effect on the Lyapunov exponents are also small. The solution is to select carefully not only the location of the balls but also their shape. This is done using a special adapted Riemannian metric such that on the one hand domination and hyperbolicity are seen in a single iterate (as in [23]), and on the other, for a large proportion of points with respect to the reference measure m, the rates of expansion on a single iterate with respect to the adapted metric are very close to the Lyapunov exponents. We only perform the perturbations on balls around those good points. In this way we can ensure that the effect of the perturbation on the Lyapunov exponents is considerable. In this regard, we also remark that we have no bound for the C 1 size of the deformation pf t q tPr0,1s that we eventually construct in Theorem 1.5; therefore we need special care with the quantifiers: we must ensure some effectiveness of the perturbation without knowing which diffeomorphism we are perturbing.
Concerning the second difficulty, we note that the type of perturbations sketched above always has some small "noisy" effect on the Lyapunov exponents that cannot be made exactly zero (except if some of the invariant subbundles are smoothly integrable). To resolve this, we define our perturbations depending on several parameters, allowing us to move the Lyapunov spectrum in all directions. By topological reasons, this eventually permits us to obtain open sets of spectra. Now, in order to construct these multiparametric perturbations, in principle one could try to compose several Baraviera-Bonatti-like perturbations that mix different pairs of Lyapunov exponents. This idea turns out to be impractical, essentially because one would need to consider adapted metrics and towers depending on parameters. The best solution is to define a new type of multi-parametric model 6 Furthermore, the main result of Baraviera and Bonatti [6] also applies to non-ergodic diffeomorphisms, in which case it controls only averaged Lyapunov exponents. Avila, Crovisier, and Wilkinson [5] perfected the method in order to control pointwise Lyapunov exponents directly without assuming ergodicity. These subtleties do not concern us because our dynamics are ergodic.
perturbation that includes Baraviera-Bonatti perturbations as a particular case. So the linear algebra gets somewhat more involved. Now let us outline the proof of Theorem 1.7. In order to manipulate, say, the first two Lyapunov exponents while keeping the third one unchanged we follow the same strategy as in the proof of Theorem 1.5, but using Baraviera-Bonatti perturbations that preserve the center-unstable foliation. This is possible because we start with the automorphism F L for which this foliation is smooth. In the converse direction, suppose that f is a conservative Anosov diffeomorphism of T 3 homotopic to F L whose Lyapunov spectrum is simple but is not majorized by the spectrum of L. For example, consider the case u " 2 and λ 1 pf q ą λ 1 pLq. If f had a simple dominated splitting then the exponential growth rate of the strong unstable foliation of f would be bigger than the corresponding rate for F L , and this would contradict the quasi-isometric property of this foliation, obtained by Brin, Burago, and Ivanov [15].

1.7.
Organization of the paper. The rest of this paper is organized as follows. In Section 2 we state the technical Proposition 2.1, which is a local multi-parametric version of Theorem 1.5, and we show how it implies the theorem. In Section 3 we construct the adapted metrics mentioned above. In Section 4 we use them to define damping perturbations; these are actually "large perturbations", but we show that their results are still Anosov with simple dominated splitting if return times to the support of the perturbation are sufficiently large. In Section 5 we define the local model of our multiparametric perturbations, and perform some computations concerning those. In Section 6 we use to the results of the previous sections and a tower construction to prove Proposition 2.1. In Section 7 we prove Theorem 1.7. In the final Section 8 we discuss extensions of our methods and other directions for the flexibility program.

Reduction to a central proposition
Given an ergodic f P Diff r m pM q, recall that λpf q denotes the Lyapunov spectrum of f with respect to the volume measure m. We defineλpf q :" T pλpf qq, that is,λpf q is the vector whose j-th entryλ j pf q is the sum of the j biggest Lyapunov exponents. This is a natural object to consider becausê λ j pf q equals the top Lyapunov exponent of the j-fold exterior power of the derivative cocycle. It is also convenient to defineλ 0 pf q :" 0 -λ d pf q. The fact that λ 1 pf q ě¨¨¨ě λ d pf q means that the function j P t0, . . . , du Þ Ñλ j pf q is concave: see Fig. 1. In this section we state Proposition 2.1, which roughly says that we can perturb f in order to slightly lower the graph of j P t0, . . . , du Þ Ñλ j pf q, and that different vertices of the graph can be moved somewhat independently. We also show how Proposition 2.1 implies the main theorem (Theorem 1.5).
Given u P t1, 2, . . . , d´1u, define a "gap function" g u : R d Ñ R by: Note that if f is a conservative Anosov diffeomorphism of unstable index u and admitting a simple dominated splitting then g u pλpf qq is strictly positive.
Proposition 2.1 (Central proposition). Let u P t1, 2, . . . , d´1u, and let a 1 , . . . , a d´1 , σ, and δ 0 be positive numbers. Then there exists δ P p0, δ 0 q so that the following holds: If f P Diff r m pM q is a conservative Anosov diffeomorphism of unstable index u, admitting a simple dominated splitting, and such that g u pλpf qq ě σ, then there exists a continuous map t P r0, 1s d´1 Þ Ñ f t P Diff r m pM q where f p0,...,0q " f and for each t " pt 1 , . . . , t d´1 q P r0, 1s d´1 , the conservative diffeomorphism f t is Anosov of unstable index u, admits a simple dominated splitting, and, for each j P t1, . . . , d´1u, The following consequence is intuitively obvious (see Fig. 2): Corollary 2.2. In Proposition 2.1, the set Λ :" tλpf t q ; t P r0, 1s d´1 u satisfies: Proof of Corollary 2.2. The second inclusion comes from (2.3). For the first one, we need the following topological fact: be a continuous map such that for every z " pz 1 , . . . , z d´1 q P r´1, 1s d´1 and every j P t1, . . . , d´1u we have: Then the image of ϕ contains the cube r´1 3 , 1 3 s d´1 .
Proof. Let C :" r´1, 1s d´1 . Note that: (2.6) @ z P BC, the segment rz, ϕpzqs does not intersect the cube 1 3 C. Consider the map ψ : C Ñ R d´1 that coincides with a rescaled version of ϕ on the subcube 1 2 C, and on the remaining shell interpolates linearly between ϕ| BC and id| BC . More precisely, letting }¨} 8 denote the maximum norm in R d´1 (whose unit closed ball is the cube C), we define: In particular, ψ coincides with the identity on the boundary BC.
By contradiction, suppose that 1 3 C ϕpCq contains a point w. Then it follows from observation (2.6) that the image of ψ does not contain w. Fix a retraction π of R d´1 twu onto BC. Then π˝ψ is a retraction of the cube C onto its boundary. It is a known fact that no such map exists. This contradiction completes the proof of the lemma. Now Corollary 2.2 follows immediately by an affine change of coordinates; namely we apply Lemma 2.3 to: ϕpzq :" B´λ`f Apzq˘w here A " pA 1 , . . . , A d´1 q, B " pB 1 , . . . , B d´1 q are the maps: Proof of Theorem 1.5. Let f and ξ be as in the statement of the theorem. Let σ :" 1 2 min g u pλpf qq, g u pξq ( . Recalling (2.1), letξ :" T pξq and pa 1 , . . . , a d´1 q :"λpf q´ξ. Since λpf q strictly majorizes ξ, each a j is positive. The function g u˝T´1 is ě 2σ on the segment rξ,λpf qs. Fix a small positive δ 0 ă 1 such that the function g u˝T´1 is ě σ on the following neighborhood of the segment: Let δ " δpa 1 , . . . , a d´1 , σ, δ 0 q be given by Proposition 2.1. Let n :" tδ´1u; this is a positive integer since 0 ă δ ă δ 0 ă 1. For each i P t0, 1, . . . , nu, let: Note that 1 n P rδ, 2δs. Therefore, for each i P t0, . . . , n´1u, We now construct a continuous path pf s q sPr0,1s of conservative Anosov diffeomorphisms as follows. Applying Proposition 2.1 to the diffeomorphism f 0 :" f , we obtain a certain family of Anosov diffeomorphisms pg 0,t q, where t runs in the cube r0, 1s d´1 . Sinceλpf 0 q " η 0 , by Corollary 2.2 and (2.7), there exists t 0 in the cube such thatλpg 0,t 0 q " η 1 . Let f 1{n :" g 0,t 0 , and define f s for s in the interval r0, 1{ns by f s :" g 0,nt 0 s . Note that these diffeomorphisms obey the gap condition g u pλpf s qq ě σ. We continue recursively in the obvious way: we apply Proposition 2.1 and Corollary 2.2 to the diffeomorphism f 1{n , extend the family f t to the interval r1{n, 2{ns and so on. We end up defining a path pf s q sPr0,1s of conservative Anosov diffeomorphisms of unstable index u admitting simple dominated splittings, such thatλpf i{n q " η i for each i P t0, . . . , nu. In particular,λpf 1 q "ξ, or equivalently, λpf 1 q " ξ, as desired.
Remark 2.4 (Prescribing paths of spectra). The proof of Theorem 1.5 can be easily adapted so that the path pλpf t qq tPr0,1s is C 0 -close to any given path from λpf q to ξ that satisfies property (a) and is monotone with respect to the partial order given by majorization.
In Sections 3 to 5 we establish several preliminary results which will be eventually used to prove the central proposition (Proposition 2.1) in Section 6.

Lyapunov metrics and charts
3.1. Lyapunov metrics. There are different ways to introduce Riemannian metrics that are suitable to study particular classes of hyperbolic dynamical systems. Such metrics are usually called Lyapunov or adapted metrics and both terms are used in more than one meaning. For various versions of such Lyapunov metrics, see e.g. [7,23,27]. Here we will construct a variant which specifically fits our setting.
Suppose that f P Diff r m pM q is a conservative Anosov diffeomorphism admitting a simple dominated splitting T M " E 1 '¨¨¨' E d . Given a C 0 Riemannian metric xx¨,¨yy, let |||¨||| denote the induced vector norm, and consider the following expansion functions χ 1 , . . . , χ d : M Ñ R: It is clear that each χ j is continuous and its integral is λ j pf q. We say that xx¨,¨yy is an Lyapunov metric if: ‚ the bundles E j are mutually orthogonal with respect to xx¨,¨yy; ‚ the expansion functions satisfy, for every x P M , where u is the unstable index of f . Proposition 3.1 (Lyapunov metric with L 1 estimate). Let f P Diff r m pM q be a conservative Anosov diffeomorphism admitting a simple dominated splitting. Furthermore, let ε ą 0. Then there exists a Lyapunov metric such that each expansion function is L 1 -close to a constant: We could smoothen the metrics given by the proposition, but in that case we would lose the orthogonality of the bundles of the dominated splitting, which is convenient for our later calculations.
Proof of Proposition 3.1. We fix an arbitrary Riemannian metric on M . Consider the following continuous functions: For each j, the sequence above forms an additive cocycle with respect to the dynamics f . By domination and hyperbolicity, if N is sufficiently large then: On the other hand, the functions θ pnq j {n converge m-almost everywhere to the constant λ j pf q. So, increasing N if necessary, we assume that: Now, for each x P M , j P t1, . . . , du, and v P E j pxq, we let: This defines a norm on each one-dimensional bundle E j pxq. Consider the unique inner product on T x M that makes those bundles orthogonal and whose induced norm agrees with the definition above.
If v P E j pxq is nonzero then, by telescopic multiplication, This means that the expansion function χ j defined by (3.1) equals θ Remark 3.2. It follows from the proof that the Lyapunov metrics can be constructed so that the tempering property |χ j˝f´χj | ă ε also holds. Remark 3.3. Our proof used that the bundles of the dominated splitting are one-dimensional, so that the geometric average defined by the formula (3.6) defines a norm. If the bundles E j are higher dimensional, one can still adapt this trick by taking an appropriate notion of averaging on the symmetric space of inner products: see [11, p. 1839].

Lyapunov charts.
Proposition 3.4 (Lyapunov charts). Suppose that f P Diff r m pM q is a conservative Anosov diffeomorphism of unstable index u admitting a simple dominated splitting T M " E 1 '¨¨¨' E d , and that xx¨,¨yy is a Lyapunov metric.
Then for all (c) Φ x has constant jacobian, that is, the push-forward of Lebesgue measure on B 0 equals the restriction of the measure m to Φ x pB 0 q times a constant factor; (d) the derivative L x :" DΦ x p0q takes the canonical basis te 1 , . . . , e d u of R d to a basis tL x pe 1 q, . . . , L x pe d qu of T x M which is orthonormal for the Lyapunov metric xx¨,¨yy x , and moreover L x pe j q P E j pxq for each j. Furthermore, tΦ x ; x P M u is a relatively compact subset of C 8 pB 0 , M q.
Proof. As it is well-known and elementary (see e.g. [29, p. 6]), the manifold M admits a conservative atlas, that is, an atlas whose charts F i : U i Ă R d Ñ M are such that the push-forward of Lebesgue measure on U i coincides with the measure m restricted to F i pU i q. By compactness, we assume that this atlas is finite. Now, given x P M , let L x : R d Ñ T x M be a linear isomorphism that sends the coordinate axes to the bundles E 1 , . . . , E d , and such that the push-forward of the standard inner product is the Lyapunov metric xx¨,¨yy x . Choose i such that F i pU i q Q x, and define: for every z in a sufficiently small closed ball B 0 Ă R d around 0. It is clear that these maps have the asserted properties (a)-(d). Moreover, all their derivatives are uniformly bounded, so we obtain relative compactness of the family.
The maps Φ x given by the proposition are called Lyapunov charts. Note that the derivative of f at an arbitrary point x P M can be diagonalized using the Lyapunov charts as follows:

Damping perturbations
In this section we define a certain type of perturbationsf of an Anosov diffeomorphism f , called damping perturbations. The idea comes from Baraviera and Bonatti [6].
Damping perturbations are actually "large perturbations", in the sense that the C 1 or even the C 0 distance betweenf and f may be large. On the other hand, the support of the perturbation, i.e. the set Z :" tx ;f pxq ‰ f pxqu is required to be "dynamically small": the return times from Z to itself are not smaller than a large number N . On Z we impose a transversality condition: essentially we want the spaces Df pxqE u pxq and Df pxqE s pxq to be transverse to E s pf pxqq and E u pf pxqq, respectively, where E u 'E s denotes the hyperbolic splitting of f . We show that if the least return time N is large enough then the perturbed diffeomorphismf is still Anosov. Moreover, on the set Z the new unstable bundleẼ u is close to the old one E u , while on the set f pZq they can be far apart. As we go upwards in the tower with base Z, the bundleẼ u is attracted by E u , so it suffers some "damping" before returning to Z and getting "kicked" again. See Fig. 3. Actually we work with Anosov diffeomorphisms with simple dominated splitting; we show that domination also persists under damping perturbations. Let us give precise statements. For each x P M , let be a linear map that takes the canonical basis te 1 , . . . , e d u of R d to a basis tL x pe 1 q, . . . , L x pe d qu of T x M that is orthonormal for the Lyapunov metric xx¨,¨yy x , and moreover L x pe j q P E j pxq for each j. Note that the map L x is unique modulo composing from the right by a matrix of the form diagp˘1, . . . ,˘1q. Taking quotient by this finite group, L x becomes unique and continuous. For each j P t1, . . . , d´1u and τ ą 0, we define the standard horizontal and vertical cones of index j and opening τ as the following subsets of euclidian space R d : Then we define the following continuous fields of cones on the tangent bundle T M : H j px, τ q :" L x pH j pτ qq , V j px, τ q :" L x pV j pτ qq .
By (3.7), we have the following invariance properties, for every τ ą 0,: Df pxqH j px, τ q Ď H j`f pxq, e´r χ j pxq´χ j`1 pxqs τ˘, (4.4) Df´1pxqV j px, τ q Ď V j`f´1 pxq, e´r χ j pf´1pxqq´χ j`1 pf´1pxqqs τ˘. Let P u and P s : T M Ñ T M be the projections respectively on the unstable and stable bundles, so that their sum is the identity.
Fix numbers α ą β ą 0, κ ą 0, σ ą 0, and N ě 2. We say that a diffeomorphismf P Diff r m pM q is a pα, β, κ, σ, N q-damping perturbation of f with respect to the metric xx¨,¨yy if the following conditions hold: (i) for all x P M and j P t1, . . . , d´1u, we have: the sets Z, f pZq, . . . , f N´1 pZq have disjoint closures. (iv) For all x P Z we have, in terms of notation (2.2): Note that the definition is symmetric under time-reversal, i.e.f´1 is a pα, β, κ, σ, N q-damping perturbation of f´1. Indeed, tx P M ;f´1pxq ‰ f´1pxq or Df´1pxq ‰ Df´1pxqu " f pZq.
Remark 4.1. Note that condition (iv) is weaker than the condition g u˝χ ě σ 2 , which is always satisfied in this paper. The reason why we insist in working with the more complicated condition (iv) is that it may be useful for future applications of our methods; moreover working with the stronger condition would not make the proof of Proposition 4.2 below any simpler.

4.2.
Properties of damping perturbations. We will show that damping perturbations are still Anosov with simple dominated splitting, provided the parameter N is large enough. It is important that the condition on N depends only on the other parameters, and not on f itself.
Let f P Diff r m pM q be a conservative Anosov diffeomorphism of unstable index u admitting a simple dominated splitting T M " E 1 '¨¨¨' E d , and fix a Lyapunov metric for f . Proposition 4.2. Let N ě N 0 and letf P Diff r m pM q be a pα, β, κ, σ, N qdamping perturbation of f with respect to the Lyapunov metric.
Thenf is a conservative Anosov diffeomorphism of unstable index u, and it admits a simple dominated splittingẼ 1 '¨¨¨'Ẽ d .

Let:
A :" Then A is an open set and A Y Z " M . Let ρ 1`ρ2 " 1 be a partition of unity subordinated to this open covering. Define the function ω j as follows: n"1 f n pZq then we let ω j pxq :" γ; ‚ if x P f pZq "f pZq then we let: ω j pxq :" γρ 1 pf´1pxqq`αρ 2 pf´1pxqq .
‚ if x P f n pZq where 2 ď n ď N´1 then, assuming ω j was already defined on f n´1 pZq, we let: The function ω j is continuous and has properties (a) and (b). Next we check property (c) case by case. First consider x P Z.
‚ If x P Z X A then ω j pxq " γ and ω j pf pxqq ě γ, so (c) follows directly from the definition of A.
‚ If x P Z A then ω j pxq " γ ă β and ω j pf pxqq " α, so (c) follows from condition (i) of the definition of damping perturbations.
Next we consider the cases where x R Z and sof pxq " f pxq and Df pxq " Df pxq.
‚ Finally, if x P f N´1 pZq then, letting y :" f´N`2pxq, we have: Using the fact that ω j pyq ď α, condition (iv) of the definition of damping perturbations, and the choice of ε, we obtain that ω j pxq " γ. Of course, ω j pf pxqq also equals γ, so (c) again follows from (4.4).
This completes the proof of the lemma.
Property (c) from Lemma 4.3 means that for each j, the continuous cone field C j pxq :" H j px, ω j pxqq is strictly forward invariant. This implies that the diffeomorphismf has a simple dominated splittingẼ 1 '¨¨¨'Ẽ d such that for all x and j, we haveẼ 1 pxq '¨¨¨'Ẽ j pxq Ă C j pxq; see e.g. [40, Proposition 2.2]. In particular, conclusion (a) of Proposition 4.2 is satisfied. Since the definition of damping perturbations is symmetric under time-reversal, it follows that conclusion (b) is also satisfied.
Next, consider the subbundles: Let us check that the Df -invariant splittingẼ u 'Ẽ s is uniformly hyperbolic. Recall that P u : T M Ñ T M denotes the projection onto E u with kernel E s . Then, for every x P M and v P T x M , (4.8) |||P u Df pxq v||| ě e χupxq |||P u v||| ě a 1 |||P u v||| , for some constant a 1 ą 1. Now consider x P Z and v P C u pxq; then: Let a 2κe pN´1qσ{2 , which by inequality (4.7) is bigger than 1. Let a 3 :" minta 1 , a 1{N 2 u ą 1. We claim that there exists a constant c ą 0 such that for all x P M , v P C u pxq, and n ě 0, (4.9) |||P u Df n pxq v||| ě ca n 3 |||P u v||| . Indeed, the inequality holds with c " 1 if the segment of orbit tx,f pxq, . . . ,f n´1 pxqu: ‚ either does not intersect Z; ‚ or, for each time it enters Z, it goes through the tower Z \ f pZq \¨¨\ f N´1 pZq completely. In general the segment may end inside the tower, in this case its contribution is uniformly bounded, so we conclude that (4.9) holds for an appropriate value of c.

The model deformation
In this section we define a special family of diffeomorphisms called "model deformation" that will be the basis for the construction in Section 6. We also establish a few properties of those maps.
Let }¨} denote the euclidian norm in R d , and let B :" tz ; }z} ď 1u be the unit ball. Denote by m the Lebesgue measure on R d . All the constructions in this section are in R d , so there is no risk of confusion with the volume measure on the manifold M .
Let Diff 8 m pB, BBq denote the set of all maps h : B Ñ B that can be extended to volume-preserving C 8 -diffeomorphisms of R d that coincide with the identity outside B.
For each j P t1, . . . , d´1u and θ P R, let R pjq θ be the orthogonal matrix that rotates the plane t0u j´1ˆR2ˆt 0u d´j´1 by angle θ and is the identity on its orthogonal complement, that is, Fix a rotationally symmetric (i.e. only depending on the norm) C 8function ρ : R d Ñ R which is not identically zero and is supported on B. For each j P t1, . . . , d´1u and s P R, define h For each t " pt 1 , . . . , t d´1 q P R d´1 , define h t P Diff 8 m pB, BBq by This d´1-parameter family of diffeomorphisms will be called the model deformation.
If t P R d´1 is sufficiently close to p0, . . . , 0q then the diffeomorphism h t is sufficiently C 1 -close to the identity so that the following transversality condition holds: for all z P B and j P t1, . . . , d´1u Dh t pzqpR jˆt 0u d´j q is transverse to t0u jˆRd´j and (5.4) Adjusting ρ if necessary, we assume from now on that every t in the unit cube r0, 1s d´1 satisfies the transversality condition.
If A is any dˆd matrix and I, J Ď t1, . . . , du are nonempty subsets of the same cardinality, let us denote by ArI, Js the submatrix formed by the entries with row in I and column in J. Recall that the determinants of those matrices are called the minors of A; the k-th principal minor corresponds to I " J " t1, . . . , ku, and is denoted by det k A.
Note that the derivatives of the maps (5.2) have the following form: Moreover, since these matrices have unit determinant, we conclude that their principal minors are: where ∆ pjq s pzq is defined as the pj, jq-entry of the matrix Dh pjq s pzq, i.e. the circled entry above.
The matrices Dh pjq s pzq have no common block triangular form, so the derivative of the composition (5.3) is intricate. Nevertheless, there is a simple expression for its principal minors: Lemma 5.1. For all z P B, t " pt 1 , . . . , t d´1 q P R d´1 , and j P t1, . . . , d´1u we have: Proof. Fix z and t, and let P :" Dh t pzq. For each k P t0, . . . , d´1u, let z pkq :" h pk´1q t k´1˝¨¨¨˝h p1q t 1 pzq and if k ą 0 let A k :" Dh pkq t k pz pk´1q q. By the chain rule, P " A d´1¨¨¨A2 A 1 .
Fix j P t1, . . . , d´1u, and let J :" t1, . . . , ju. We claim that if I Ă t1, . . . , du has cardinality j and I ‰ J then: det A k rI, Js " 0 for every k ă j, and (5.7) det A k rJ, Is " 0 for every k ą j. (5.8) Indeed, for all i ą j, the row matrix A k rtiu, Js vanishes if k ă j, and the column matrix A k rJ, tius vanishes if k ą j.
By the Cauchy-Binet formula for the minors of a product, where the sum is over all d´2-tuples pI 1 , . . . , I d´2 q of subsets of t1, . . . , du of cardinality j. Consider a nonzero term of this sum. Using (5.7) recursively we obtain J " I 1 " I 2 "¨¨¨" I j´1 . On the other hand, using (5.8) recursively we obtain J " I d´2 " I d´3 "¨¨¨" I j . Therefore the sum contains a single nonzero term, which by (5.6) equals ∆ pjq t j pz j´1 q.
The transversality condition (5.4) is equivalent to the non-vanishing of the minors det j Dh t pzq. When t " 0 these minors equal 1. So the minors are always strictly positive. In particular, by Lemma 5.1, the functions ∆ pjq s are strictly positive for s P r0, 1s. We will need information about the logarithms of these functions. By definition, ∆ pjq s pz 1 , . . . , z d´1 q " B Bz j´z j cospsρpzqq´z j`1 sinpsρpzqq¯.
Let us discuss a few other features of the model deformation.
In fact the inclusions are equivalent to one another since the complement of a horizontal cone H j pτ q is (modulo sets of non-empty interior) a vertical cone V j pτ q. Let us fix these numbers α ą β ą 0 from now on. Let P j and Q j : R d Ñ R d be the projections on the spaces R jˆt 0u d´j and t0u jˆRd´j , respectively. We fix κ ą 0 such that (5.12) " v P H j pβq ñ }P j Dh t pzq v} ě κ }P j v} , v P V j pβq ñ }Q j Dh´1 t pzq v} ě κ }Q j v} .
Let us recall a few facts about exterior powers. Let te 1 , . . . , e d u denote the canonical basis in R d . Then, for each k P t1, . . . , du, Λ k R d is a vector space with the canonical basis te i 1^¨¨¨^e i k u i 1 ă¨¨¨ăi k . We endow Λ k R d with the inner product that makes the canonical basis orthonormal. There is a wedge operation pv, wq P pΛ k R d qˆpΛ ℓ R d q Þ Ñ v^w P Λ k`ℓ R d which is associative, mutilinear, and skew-symmetric (i.e., w^v " p´1q kℓ v^w). Given a linear map A : The entries of the matrix of the exterior power Λ k A with respect to the canonical basis are exactly the kˆk minors of A; this is called the k-th compound matrix of A. More precisely, if I " ti 1 ă¨¨¨ă i k u and J " tj 1 ă¨¨¨ă j k u then A e i 1^¨¨¨^e i k , pΛ k Aqpe j 1^¨¨¨^e j k q E " det ArI, Js .
So the Cauchy-Binet formula used before is nothing but the functoriality of the exterior powers, i.e., Λ k pABq " pΛ k AqpΛ k Bq. If A is an orthogonal linear map, then so is Λ k A. Therefore the inner product on Λ k R d depends only on the inner product on R d , and not on the choice of orthonormal basis; equivalently, the k-fold exterior power of an inner product space is an inner product space. From a more geometrical viewpoint, if v 1 , . . . , v k are vectors in R d then the norm of v 1^¨¨¨^vk is the k-dimensional volume of the parallelepiped spanned by these vectors.
Coming back to the model deformation, let us note for later use that for any ν ą 0 there exists γ P p0, βq such that for all z P B, t P r0, 1s d´1 , j P t1, . . . , d´1u and all linearly independent vectors w 1 , . . . , w j P H j pγq, we have (5.13)ˇˇˇˇl og @ pΛ j Dh t pzqqpw 1^¨¨¨^wj q, e 1^¨¨¨^ej D xw 1^¨¨¨^wj , e 1^¨¨¨^ej y´l og det j Dh t pzqˇˇˇˇă ν ; this statement includes the fact that the numerator and the denominator in the expression above are both nonzero and have the same sign.
We fix several other constants. Let α ą β ą 0 be openings with property (5.11). Let κ ą 0 be a constant with property (5.12). Let ν :" pδ 0 {2q minta 1 , . . . , a d´1 u and take γ P p0, βq with property (5.13). Let N :" N 0 pα, β, γ, κ, σq be given by (4.6). Let δ :" δ 0 {N ; this will be the scaling factor as in the statement of Proposition 2.1. Choose a very small ε ą 0; this choice will be apparent at the end of the proof. Now let us pick f P Diff r m pM q as in the statement of Proposition 2.1, that is, a conservative Anosov C r -diffeomorphism of unstable index u, admitting a simple dominated splitting, and such that g u pλpf qq ě σ.
We apply Proposition 3.1 and find a Lyapunov metric so that the associated expansion functions χ 1 , . . . , χ d obey the L 1 -estimate (3.3) with the chosen value of ε. Consider the sets R j :" x P M ; |χ j pxq´λ j pf q| ě σ{2 ( . By a trivial estimate (Markov's inequality), mpR j q ď 2σ´1ε. So the following open set: has measure mpU q ą 1´4dN σ´1ε. By the Rokhlin Lemma, there exists a measurable set Z 1 disjoint from f pZ 1 q, f 2 pZ 1 q, . . . , f N´1 pZ 1 q with measure Then Let Z 2 be a compact subset of Z 1 X U satisfying the same bound.
Then take an open neighborhood Z 3 Ď U of Z 2 that satisfies the same bound, and moreover is disjoint from f pZ 3 q, f 2 pZ 3 q, . . . , f N´1 pZ 3 q.
Let Φ x : B 0 Ñ M be the Lyapunov charts coming from Proposition 3.4; here B 0 is a rescaling of the unit closed ball B, that is B 0 " s 0 B for some s 0 ą 0. For every x P M and every s P p0, s 0 s, let us define a Lyapunov ball as: Reducing s 0 if necessary, we assume that the following properties hold: y, y 1 P Bpx, sq ñ |χ j pyq´χ j py 1 q| ă ε , (6.2) where L y :" DΦ y p0q.
Since the Lyapunov charts Φ x form a relatively compact subset of C 1 pB 0 , M q, the family of Lyapunov balls contained in the open set Z 3 forms a Vitali covering of Z 3 . Therefore there exists a finite collection of disjoint Lyapunov balls Bpx 1 , s 1 q, . . . , Bpx p , s p q Ď Z 3 whose union Z 4 has measure mpZ 4 q ą mpZ 3 q´ε. To simplify notation, let B i :" Bpx i , s i q and define (6.4) Ψ i : B Ñ M by Ψ i pzq :" Φ x i ps i zq ; so Ψ i has constant jacobian and image B i .
For each t P r0, 1s d´1 , let us define g t P Diff 8 m pM q as follows: g t equals the identity outside of Z 4 , and on each B i it is defined as: where Bpt 1 , . . . , t d´1 q :" pb 1 t 1 , . . . , b d´1 t d´1 q, and b j comes from (6.1). The sought-after deformation of f is defined as: We now need to check that the maps f t have the properties asserted in Proposition 2.1. As a first step, we show: Proof. Condition (i) in the definition of damping perturbations follows from the inclusions (5.11) and the fact that Df (resp. Df´1) decreases the opening of horizontal (resp. vertical) cones: see (4.4), (4.5). Let us check condition (ii). If v P H u px, βq then, using property (5.12), while if v P V u px, βq then Df´1pxq v P V u pf´1pxq, βq and so: Condition (iii) follows from the fact that the set Z 4 is disjoint from its N´1 first iterates, while condition (iv) follows from the fact that Z 4 is contained in U . Now Proposition 4.2 ensures that each f t is an Anosov diffeomorphism of unstable index u and admits a simple dominated splitting T M " E 1,t '¨¨' E d,t . Moreover, by part (a) of the proposition, The rest of the proof consists in estimating the summed Lyapunov exponentsλ j pf t q.
For each x P M and j P t1, . . . , du, let v j pxq :" L x e j , which is a vector that spans E j pxq. Since these vectors form an orthonormal basis of T x M , using the definition (3.1) of the expansion function χ j , we obtain that for every vector w P T x M , (6.8) xxDf pxqw, v j pf pxqqyy " e χ j pxq xxw, v j pxqyy .
In each exterior power of the tangent bundle we consider the corresponding exterior power of the Lyapunov metric, denoting it by the same symbol. Letv j pxq :" v 1 pxq^¨¨¨^v j pxq P Λ j T x M . Note that for everyŵ P Λ j T x M , (6.9) xxΛ j Df pxqŵ,v j pf pxqqyy " e χ 1 pxq`¨¨¨`χ j pxq xxŵ,v j pxqyy .
For each x P M , j P t1, . . . , du, and t P r0, 1s d´1 , let us choose a vector v j,t pxq that spans E j,t pxq in such a way that it depends continuously on t and equals v j pxq when t " 0. Let v j,t pxq :" v 1,t pxq^¨¨¨^v j,t pxq P Λ j T x M .
Note that the numerator and the denominator in this formula are both positive, thanks to (6.6).
Proof. The top Lyapunov exponent of the linear cocycle Λ j Df t equalsλ j pf t q, has multiplicity 1, and the corresponding Oseledets space at a regular point x is spanned byv j,t pxq (see [3,Theorem 5.3.1]). In particular, the corresponding expansion function log |||Λ j Df t pxqv j,t pxq||| |||v j,t pxq||| logψ j,t pxq has integralλ j pf t q. Furthermore, we have: Λ j Df t pzqv j,t pxq "˘ψ j,t pxqv j,t pf t pxqq .
Substituting into (6.10), we see that the functions log ψ j,t and logψ j,t are cohomologous. In particular they have the same integral, proving the lemma.
Let us analyze the function log ψ j,t in order to estimate its integral. Lemma 6.3. If x R Z 4 then log ψ j,t pxq " χ 1 pxq`¨¨¨`χ j pxq.
Proof. If x R Z 4 then f t " f on a neighborhood of x and so the lemma follows from (6.9).
On the other hand, on Z 4 " Ů i B i we have the following estimate: Lemma 6.4. If x P B i and z :" Ψ´1 i pxq then: (6.11)ˇˇlog ψ j,t pxq´"χ 1 pxq`¨¨¨`χ j pxq`log det j Dh Bt pzq ‰ˇˇˇď ν`Opεq .
Here and in what follows, Opεq stands for anything whose absolute value is bounded by ε times something depending on the numbers fixed at the beginning of the proof and on the model deformation.
Combining this with (6.14) and (6.12), we obtain the desired estimate (6.11) for the case j " 1.
The proof for j ě 2 follows exactly the same pattern, taking exterior powers, of course. The only point that deserves notice is that estimate (6.13) should be replaced by the following: if e j,t :" L´1 x v j,t pxq then |xe 1,t^¨¨¨^ej,t , e 1^¨¨¨^ej y| ě p1`γ 2 q´j {2 }e 1,t^¨¨¨^ej,t } .
Indeed, the orthogonal projection onto the space spanned by e 1 , . . . , e j cannot contract a vector in the cone H j pγq by a factor smaller than p1`γ 2 q´1 {2 , and hence it cannot contract the volume of a j-dimensional parallelepipid in the cone by a factor smaller than p1`γ 2 q´j {2 .
We obtain from (6.11) and (5.10) that for each Lyapunov ball B i ,ˇˇˇż This together with Lemmas 6.2 and 6.3 yields:ˇλ j pf t q´λ j pf q`mpZ 4 qQpb j t j qˇˇď mpZ 4 qν`Opεq . Since ν ď δ 0 a j {2 and 1{N´mpZ 4 q " Opεq, we have:ˇˇˇλ By the definition (6.1) of b j we have: Combining these two pieces of information, and recalling that δ " δ 0 {N , we obtain:λ j pf t q´λ j pf q ě

Proof of additional results for tori
In this section we prove two supplements to Theorem 1.5, namely Corollary 1.6 and Theorem 1.7. 7.1. Anosov diffeomorphisms display all hyperbolic simple Lyapunov spectra. In order to deduce Corollary 1.6 from Theorem 1.5, we need the following fact: Proof. We need to show the existence of a polynomial P pxq with leading term x d and constant term˘1 whose roots are all real and simple, being u of them with modulus bigger than 1 and d´u of them of modulus smaller than 1. Then the companion matrix of P pxq will be an element L P GLpd, Zq that induces an Anosov linear automorphism F L : T d Ñ T d with unstable index u and simple Lyapunov spectrum. Though the existence of such polynomials can be quickly deduced from Dirichlet's unit theorem, we will provide a completely elementary proof. The idea comes from a proof of existence of Pisot-Vijayaraghavan numbers of arbitrary degree [42,Theorem 1].
Fix integers a 1 ą¨¨¨ą a u ą 0 ą a u`1 ą¨¨¨ą a d whose sum is 0 and such that a i´ai`1 ě 2 for each i P t1, . . . , d´1u. Let b ě 3 be another integer, and consider the polynomial Note thatâ i ě 0 for each i and so P has integer coefficients. Furthermore, P is monic and has constant term p´1q d .
Let us locate the roots of P . We claim that, for all n P Z ta 1 , . . . , a d u, (7.1) P pb n q is non-zero and has the same sign as d ź j"1 pn´a j q , and so, by the intermediate value theorem, P has d´u simple roots on the interval p0, 1q and u simple roots on the interval p1,`8q. In order to prove the claim, fix n and consider the expression: The function i P t0, . . . , du Þ Ñâ i`n pd´iq is integer-valued, strictly concave, and attains a maximum at i " k, where k is the number of negative factors in the product ś d j"1 pn´a j q. Using that 2 ř 8 j"1 b´j ď 1, we see that the term corresponding to i " k in the right-hand side of (7.2) is bigger in absolute value than the sum of all other terms. So the sign of P pb n q is p´1q k , thus proving (7.1).
Proof of Corollary 1.6. Consider nonzero numbers ξ 1 ą¨¨¨ą ξ d whose sum is equal to 0, and let ξ :" pξ 1 , . . . , ξ d q. By Lemma 7.1, there exists an Anosov linear automorphism F L : T d Ñ T d whose Lyapunov spectrum λpLq is simple and has the same unstable index as ξ. If n is sufficiently large then λpL n q " nλpLq strictly majorizes ξ. Hence by Theorem 1.5 there exists a conservative Anosov C 8 diffeomorphism f : T d Ñ T d homotopic to F L n with simple dominated splitting and such that λpf q " ξ, as we wanted to prove. 7.2. Spectra of Anosov diffeomorphisms with simple dominated splitting on T 3 . In this subsection we prove Theorem 1.7. Fix a hyperbolic matrix L P GLp3, Zq whose eigenvalues are all real and simple. So the induced automorphism F L : T 3 Ñ T 3 is Anosov and its Lyapunov spectrum is simple. Let u P t1, 2u be its unstable index.
Proof of the "only if " part of Theorem 1.7. Let f P Diff 8 m pT 3 q be an Anosov diffeomorphism homotopic to the automorphism F L , and admitting simple dominated splitting. Since f and F L are topologically conjugate, they have the same unstable index u. Taking inverses if necessary, we can assume that u " 2. So the Lyapunov spectrum λpf q " pλ 1 pf q, λ 2 pf q, λ 3 pf qq satisfies λ 1 pf q ą λ 2 pf q ą 0 ą λ 3 pf q. Let us show that λpf q is majorized by λpLq. As explained before, the inequality λ 1 pf q`λ 2 pf q ď λ 1 pLq`λ 2 pLq is immediate: see (1.3). Therefore we need to show that λ 1 pf q ď λ 1 pLq.
Letf be a lift of f to the universal covering R 3 . Since f is homotopic to F L , we havef " L`ϕ for some Z 3 -periodic map ϕ : R 3 Ñ R 3 . So, for every n ě 0,f n " L n`n´1 ÿ k"0 L k˝ϕ˝f n´1´k .
Since ϕ is bounded, it follows that there is a constant C 1 ą 0 (independent of n) such that, for all x, y P R d with }x´y} ď 1 we have: Since the top eigenvalue of the linear map L is simple, there is another constant C 2 ą 0 such that }L k } ď C 2 e kλ 1 pLq for all k ě 0. In particular, }f n pxq´f n pyq} ď C 3 e nλ 1 pLq , where C 3 ą 0 is another constant. By [15,Theorem 1.3] (see also [37,Corollary 7.7]), the strong unstable foliation in the universal covering is quasi-isometric; this means that there is a constant C 4 ą 0 such that if I Ă R 3 if a segment of strong unstable manifold then its length can be bounded as: Hence, for every such a segment with diampIq ď 1, and every n ě 0, lenpf n pIqq ď C 4 C 3 e nλ 1 pLq`C 4 . If follows from the next lemma that λ 1 pf q ď λ 1 pLq, as we wanted to prove. Lemma 7.2. For each x P T 3 , let W x Ă T 3 be the segment of strong unstable leaf of length 1 for which x is a midpoint. Then, for m-almost every x P T 3 , lim sup nÑ8 log lenpf n pW x qq n ě λ 1 pf q .
Let R be the set of points y P T 3 for which lim nÑ8 1 n log }Df n |E 1 pyq} " λ 1 pf q; then mpRq " 1. By absolute continuity of the strong unstable foliation [35,Lemma 10], for m-almost every x P T 3 we have ℓpW x X Rq " 1. Therefore: Proof of the "if " part of Theorem 1.7. Now we fix a vector ξ " pξ 1 , ξ 2 , ξ 3 q such that ξ 1 ą ξ 2 ą ξ 3 , ξ u ą 0 ą ξ u`1 , and ξ ď λpLq. We want to find a smooth conservative Anosov diffeomorphism f homotopic to F L , admitting a simple dominated splitting, and with spectrum λpf q " ξ. If ξ is strictly majorized by λpLq then the existence of f is guaranteed by Theorem 1.5. So let us assume that majorization is not strict, that is, either ξ 1 " λ 1 pLq or ξ 1`ξ2 " λ 1 pLq`λ 2 pLq. We can assume that only one of these equalities is satisfied, since otherwise we can simply take f " F L . Let E L 1 , E L 2 , E L 3 denote the eigenspaces of L corresponding to the Lyapunov exponents λ 1 pLq, λ 2 pLq, λ 3 pLq, respectively. In case ξ 1 " λ 1 pLq we shall perform the deformation in Theorem 1.5 in such a way that the foliation F 23 parallel to E L 2 'E L 3 is preserved, while in the case ξ 1`ξ2 " λ 1 pLq`λ 2 pLq (or equivalently λ 3 pf q " λ 3 pLq) we shall do it in such a way that foliation F 12 parallel to E L 1 ' E L 2 is preserved. Since both cases are dealt with similarly, we will concentrate ourselves on the second case, namely ξ 3 " λ 3 pLq.
The following observation will make the argument simpler: 3. Let f be a conservative Anosov diffeomorphism of T 3 homotopic to F L . If f preserves the foliation F 12 then λ 3 pf q " λ 3 pLq. 7 Proof. Letf be a lift of f to the universal covering R 3 . Since f is homotopic to F L , we havef " L`ϕ for some Z 3 -periodic map ϕ. Let P 1 , P 2 , P 3 be the projections associated to the splitting The fact that f preserves the foliation F 12 implies thatf preserves the foliation r F 12 of R 3 along planes parallel to E L 1 ' E L 2 . For every x P R 3 and every v P E L 1 ' E L 2 , we have P 3˝f pxq " P 3˝f px`vq and therefore P 3˝ϕ pxq " P 3˝ϕ px`vq. Since F L is a 3-dimensional Anosov automorphism, E L 2 ' E L 3 is a totally irrational plane, and since the map P 3˝ϕ is Z 3 -periodic, it must be constant. So the derivative of f written w.r.t. to the splitting Sincef is volume preserving, this implies that the absolute value of the determinant of Df restricted to E L 1 ' E L 2 is everywhere constant e´λ 3 pLq . The same is true for Df and hence λ 1 pf q`λ 2 pf q "´λ 3 pLq, that is, λ 3 pf q " λ 3 pLq.
Coming back to the proof of Theorem 1.7, we need a variation of the central proposition (Proposition 2.1) where all diffeomorphisms preserve the foliation F 12 : Proposition 7.4. Let a 1 , σ, and δ 0 be positive numbers. Then there exists δ P p0, δ 0 q with the following properties: Let f P Diff 8 m pT 3 q be an Anosov diffeomorphism homotopic to F L admitting a simple dominated splitting, and such that g u pλpf qq ě σ. In addition, assume that f preserves the foliation F 12 (and so λ 3 pf q " λ 3 pLq, by Lemma 7.3). Then there exists a continuous map t P r0, 1s Þ Ñ f t P Diff 8 m pT 3 q where f 0 " f and for each t P r0, 1s, the conservative diffeomorphism f t is Anosov, admits a simple dominated splitting, and its top Lyapunov exponent satisfies: λ 1 pf q´4δa 1 ă λ 1 pf t q ă λ 1 pf q`δa 1 , λ 1 pf q´4δa 1 ă λ 1 pf 1 q ă λ 1 pf q´2δa 1 .
In addition, each f t preserves the foliation F 12 (and so λ 3 pf t q " λ 3 pLq, by Lemma 7.3). 7 Actually, λ3,µpf q " λ3pLq for any f -invariant probability measure µ, by the same proof.
Once this is established, we mimic the proof of Theorem 1.5; namely, we concatenate paths produced by the proposition and obtain a deformation of F L that ends with a diffeomorphism having Lyapunov spectrum equal to ξ.
In order to prove Proposition 7.4, we begin by modifying the construction of the Lyapunov charts in Proposition 3.4. Since we are working in the torus T 3 " R 3 {Z 3 , we can identify tangent spaces T x T 3 with R 3 . Let π : R 3 Ñ T 3 be the quotient projection. For each x P T 3 , let L x : R 3 Ñ R 3 be a linear map that takes the canonical basis te 1 , e 2 , e 3 u of R 3 to a basis tL x pe 1 q, L x pe 2 q, L x pe 3 qu of R 3 which is orthonormal for the Lyapunov metric xx¨,¨yy x , and moreover L x pe j q P E f j pxq for each j. Then we define the Lyapunov charts Φ x pzq :" x`πpL x pzqq . These charts have all properties from Proposition 3.4 and the following additional one: (e) Φ x is a foliated chart, that is, if z 1 , z 2 , z 3 denote canonical coordinates in R 3 then Φ x maps levels set of z 3 (horizontal slices) into leaves of F 12 . Indeed, the derivatives DΦ x pzq are constant equal to L x , and L x maps the plane Then we follow the proof of Proposition 2.1 (but with a 2 " 0). To summarize, the deformation f t of f is constructed as follows: ‚ We select a disjoint family of Lyapunov balls B 1 , . . . , B p ; each ball B i equals Φ i pBq, where B is the unit ball B in R 3 and Ψ i pzq " Φ x i ps i zq is a rescaled Lyapunov chart; ‚ On each Lyapunov ball B i the deformation is defined as f t " f˝g t , where g t " Ψ i˝h p1q b 1 t˝Ψ´1 i ; here b 1 ą 0 is a constant and h p1q t is the first elementary model deformation.
Inspecting the equations (5.1), (5.2) that define the diffeomorphism h p1q t : B Ñ B, we immediately see that it preserves horizontal slices (i.e. level sets of z 3 ). Since Ψ i maps horizontal slices into leaves of F 12 , the upshot is that each diffeomorphism f t preserves the foliation F 12 . This proves Proposition 7.4. As explained before, Theorem 1.7 follows.
8. Further directions of research 8.1. Extensions of our methods. Hyperbolicity is not fundamental to our constructions; as in , domination is the important keyword. In fact, this perturbation method seems much more versatile, and should apply if domination is allowed to degenerate in a controlled way on a certain singular set, as it is the case in [25,18]. Therefore the conjectures of Subsection 1.2 seem approachable, at least in some cases.
Let us briefly discuss what happens when we near the boundary of set of allowable spectra ξ in Theorem 1.5. We consider the principal case f " F L .
Let u be the unstable index of F L . Consider the set of ξ that meet conditions (a) and (b) in the theorem. There are three types of components of the boundary: (a) ξ u " 0 or ξ u`1 " 0. One can carry our construction across either of those. The resulting map of course is not Anosov anymore but partially hyperbolic with one-dimensional central bundle. 8 (b) ř k i"1 ξ i " ř k i"1 λ i pLq for some k P t1, . . . , d´1u. Cases k " 1 and k " d´1 are feasible: as in the proof of the "if" part of Theorem 1.7, the perturbations preserve codimension one F L -invariant foliations.
For other values of k we get into difficulties. (c) ξ i " ξ i`1 for some i P t1, . . . , d´1u. Our construction degenerates since the amount of domination decreases and the Lyapunov metric explodes.
It would also be interesting to have a more explicit construction of Anosov diffeomorphisms in the torus T d (with d ě 3) with prescribed Lyapunov spectra.

8.2.
Other instances of the flexibility program.

Other measures.
In other settings, the invariant measure (or measures) one is interested in is not necessarily fixed in the class of dynamical systems under consideration, but varies with the dynamics itself. The prototypical example consists of equilibrium states of sufficiently hyperbolic dynamics with respect to relevant potentials. The flexibility paradigm then applies. See [19] for a result in this direction, where expanding maps on the circle are considered. 8.2.2. Symplectic systems. The problems that we have posed in the volumepreserving setting have symplectic counterparts, where symmetry of the spectrum appears as an extra requirement. It is plausible that our methods can be adapted to the symplectic setting, but not in a straightforward way. 8.2.3. Flows. Structural stability for flows does not imply topological conjugacy so topological entropy becomes a free parameter even in the uniformly hyperbolic case. Thus for conservative Anosov flows on three-dimensional manifolds the basic flexibility problem involves realization of arbitrary pairs of numbers as values for topological and metric entropy subject only to the variational inequality. This problem can be solved fairly easily for suspensions of T 2 and unit tangent bundles of surfaces of genus g ě 2 using time changes of homogeneous models but becomes more interesting (probably still tractable) for exotic Anosov flows where homogeneous models are not available.
The problem becomes really interesting in the standard setting when one considers special classes of Anosov flows. The prime example here is provided by geodesic flows on compact surfaces of negative curvature. In this case a natural normalization is available by fixing the total surface area. In this case the variational inequality is strengthened by the conformal inequality [26] and possible values of the metric entropy h m and topological entropy h top for Riemannian metrics of negative non-constant curvature are restricted to: where g is the genus of the surface and V is its total area. For any constant curvature metric those inequalities become equalities. The fact that equality of the metric and topological entropy implies constant curvature is a prototype case of rigidity that is discussed below. The flexibility problem in this setting is solved in [20]. The solution uses methods which are totally different from those of the present work. For flexibility results where the conformal class is fixed (and also taking other invariant quantities into account), see [8,9]. The higher dimensional case is wide open. One of the difficulties of dealing with geodesic flows on higher dimensional negatively curved manifolds is that algebraic models have non-simple Lyapunov spectrum (in fact, either one or two Lyapunov exponents of the same sign and full splittings never exist. 8.3. Flexibility and rigidity. In the context of conservative smooth dynamical systems the strongest natural equivalence relation is smooth conjugacy. For symplectic systems it is similarly a symplectic conjugacy, for geodesic flows -isometry of the underlying metrics, and so on. A general phenomenon of rigidity in this context can be described as follows: (R) Values of finitely many invariants determine the system either locally, i.e. in a certain neighborhood of a "model", or globally within an a priori defined class of systems. The space of equivalence classes at best can be given some natural infinitedimensional structure and at worst is "wild". This is proven in a number of situations but in general it has to be viewed as a paradigmatic statement, not a theorem, and, beyond C 1 case, almost every meaningful general question is open. Thus rigidity should be quite rare since it should appear for very special values of invariants.
Rigidity tends to be rare since it should appear for very special values of invariants. 9 A natural question in our context is whether particular values of Lyapunov exponents (for conservative systems, or symplectic systems, or geodesic flows) imply rigidity. In agreement with the general flexibility paradigm one may expect this to happen at the extreme allowable values of exponents. This tends to be true in low dimension, not true in full generality and probable in a number of interesting situations. Several recent results fit this pattern: see the papers [32,39,22] (concerning conservative Anosov diffeomorphisms, mostly) and [16,17] (concerning geodesic flows).