Robustly invariant sets in fibre contracting bundle flows

We provide abstract conditions which imply the existence of a robustly invariant neighbourhood of a global section of a fibre bundle flow. We then apply such a result to the bundle flow generated by an Anosov flow when the fibre is the space of jets (which are described by local manifolds). As a consequence we obtain sets of manifolds (e.g. approximations of stable manifolds) that are left invariant, {\bf for all} negative times, by the flow and its small perturbations. Finally, we show that the latter result can be used to easily fix a mistake recently uncovered in the paper {\em Smooth Anosov flows: correlation spectra and stability}, \cite{BuL}, by the present authors.


Introduction
The standard definition of Anosov map or flow states that there must be an invariant splitting that is eventually contracted or expanded [5]. It is well known that one can change the Riemannian metric so as to insure that the hyperbolicity (expansion and contraction) takes place for all times and not just after a fixed time (e.g. see [10]). Unfortunately, the new metric is usually constructed via the invariant splitting and hence has its same regularity (typically only Hölder). Also, if one considers a small perturbation of the flow or the map one has strict hyperbolicity with respect to a different metric. On the contrary, in many recent applications (e.g. the perturbation theory developed in [4] and applied to flows in [2]) it would be very convenient to have strict hyperbolicity for an open set of systems with respect to a fixed (possibly smooth) distance.
In fact, as typically one establishes hyperbolicity by finding a family of strictly invariant cones, it would be very useful to be able to specify a family of invariant cones that is invariant also for small perturbations of the map or flow and has a uniform contraction and expansion for all the vectors in the cone (also for the perturbed dynamics). Moreover, as for many applications one would like to consider sets of approximate unstable manifolds and given that such sets can be conveniently specified in the language of jets, it is tempting to state the results in terms of fibre bundles, so that a manifold of different situations would be treated all at once.
It turns out that it is fairly easy to carry out the above program in the case of discrete time. The reason being that the standard strategy provides, at time one, a fixed amount of contraction or expansion and hence a small perturbation (needed to make the metric smooth or to consider near by dynamics) will conserve such a property. Completely different is the situation for flows. The problem being the small times. For such times the amount of hyperbolicity is very small and any perturbation may destroy it. It is then not obvious if the above program can be carried out for flows or not. Here we show that it can be done. This is a consequence of the general considerations put forward in section 2. We then show in section 3 how to apply the abstract theory to the case of Anosov flows.
The problem of small times is rather ubiquitous when attempting to study fine properties of flows. In particular, in the theory developed in [1,3] it is solved by an ad hoc construction that has the draw back of using explicitly the dynamics, hence tying the theory to a fixed flow. The problem is also present, but mistakenly ignored, in [2]. Indeed, Lemma 7.2 of [2] was stated to hold for all t ≥ 0 but in fact it is only true for all t greater than some non-zero constant. This affects all the paper since [2] reduces to the study of the transfer operator, associated with the dynamics, on appropriate Banach spaces. The norms defining such Banach spaces are based on a set of admissible leaves that are required to be invariant, with respect to the dynamics, in the sense of [2,Lemma 7.2]. Since the set of leaves defined in [2] is invariant only after some finite time (contrary to the statement of [2,Lemma 7.2]) no control was obtained on the transfer operator for short time. Unfortunately, this problem cannot be fixed as in [1], since in [2] one aims at doing perturbation theory and hence one needs a Banach space (on which to study the generator of the flow) which is adapted to an open set of the flows rather than a specific one as in [1,3]. In section 4 we show that the present results yield an easy and elegant fix to the above mistake. After such a correction all the results in [2] remain true as claimed. Finally, in Appendix A we substantiate a claim of [2] that, although irrelevant for the results proved there, might have an interest in its own.

Flows on Fibre Bundles
Let (E, B, π, F ) be a fibre bundle 1 where (E, d E ) and (B, d B ) are both metric spaces and π is Lipschitz. 2 We also assume that there exists an open subset E 0 ⊂ E with compact closure and a global 3 section s : B → E 0 ⊂ E. The section s is assumed to be Lipschitz. We require that E 0 contains some neighbourhood 4 of the set s(B). Let Ψ t : E → E be a bundle flow. This is a jointly continuous 5 map (x, t) → Ψ t (x) such that Ψ 0 = Id, Ψ s • Ψ t = Ψ t+s , for each t the map Ψ t : E → E is Lipschitz, and there exists ϕ t : B → B such that ϕ t • π = π • Ψ t for all t ≥ 0. This means that the flow preserves fibres. We require the set E 0 to be eventually invariant, i.e. there exists t 0 > 0 such that Ψ t0 E 0 ⊆ E 0 . We also assume that there exists C > 0 such that, for all t > 0, 6 Finally, we are interested in the case when Ψ t is a fibre contraction in the sense that there exists σ < 1, such that 1 As usual E denotes the total space, B denotes the base space, π : E → B is a continuous surjection, and F denotes the fibre. 2 If one has the natural choice of metric in the base space defined by d B (p, p ′ ) = inf{d E (x, x ′ ) : p = πx, p ′ = πx ′ } then the Lipschitz property of π is immediate. 3 A section is a continuous map such that π • s(p) = p for all p ∈ B. Fibre bundles do not in general have global sections. 4 I.e. {x ∈ E : d E (x, s(B)) < δ} ⊂ E 0 for some δ > 0. 5 As a map E × [0, ∞) → E. 6 Note that, by the semigroup property, the invariance of E 0 and the triangle inequality, it suffices that (1) holds for t ∈ [0, t 0 ].
As we are interested in robust properties, we wish to consider an open set of flows. We say that two bundle flows Ψ,Ψ are η-close if The purpose of this section is to prove the following result concerning the construction of a robust strictly-invariant set.
Theorem 1. Suppose that Ψ t : E → E is a fibre contracting bundle flow as above. Then there exists a set K ⊂ E 0 , and η > 0 such thatΨ t K ⊆ K for all t ≥ 0, and for allΨ which are η-close to Ψ.
The remainder of this section is devoted to the proof of the above Theorem.
First note that the fibre contraction (2) and the pre-compactness of E 0 implies that there exists λ > 0, C ≥ 1 such that We would like the bundle map to be a strict fibre contraction. We therefore introduce an adapted metric. Fix some λ ′ ∈ (0, λ), t 1 > 0 sufficiently large so that Ce −(λ−λ ′ )t1 < 1 and let Proof. It suffices to prove the Lemma for t ∈ [0, t 1 ]. Using the definition of the adapted metric and the estimate (4) we have Since Ce −(λ−λ ′ )t1 < 1 this completes the proof of the lemma.
Proof. The continuity of the flow, combined with the semigroup properties implies that there exists . This estimate, along with the definition of d ′ (·, ·) suffices.
Note that the invariance of E 0 and the fibre contraction imply the existence of an invariant global section. Yet, such a section is in general only continuous (e.g. see [7, Theorem (3.1 ′ )]). Typically one expects to be able to produce smooth global sections by a smoothing of the invariant one, yet in the present generality it is not obvious how to do it. Hence our assumption on the existence of a global Lipschitz section s : B → E 0 .
Nevertheless, in the following we need to have a section close to the invariant one, while still being Lipschitz. This can be easily constructed. Let s 0 = Ψ t2 • s • ϕ −t2 : B → E for some t 2 > 0 chosen sufficiently large as specified shortly. The section s 0 is almost invariant as follows.
Lemma 4. For all ǫ > 0 there exists t 2 > 0, and C ǫ > 0, such that Proof. Let t 2 t −1 0 ∈ N and set p ′ = ϕ −t2 (p). Using the estimate of Lemma 2 The linear growth of the flow (1) and the Lipschitz continuity of π means that we have a similar linear growth estimate for ϕ t : Using the linear growth and the Lipschitz continuity of s we therefore have that The equivalence of the metrics (Lemma 3) means that a similar estimate holds for d ′ (·, ·). Choosing t 2 sufficiently large we ensure that Ce −λ ′ t2 ≤ ǫ, which proves the first inequality of the Lemma. Since, by hypotheses, B is compact and ϕ t Lipschitz, it follows by the semigroup property and the linear growth that there exists Using the latter inequality and Lemma 2 the final statement of the lemma follows. Note that it is to be expected that C ǫ becomes larger as ǫ is chosen smaller.
We are now in a position to define the strictly-invariant set. Let Proof of Theorem 1. We must show that x ∈ K implies thatΨ t x ∈ K. By the semigroup property of the flow it suffices to prove the result for all t ∈ [0, t 3 ] for some fixed, possibly small, t 3 > 0. We estimate In the last line we used (3) and Lemma 3, Lemma 2, and Lemma 4 for the first three terms respectively. The final term is a combination of (3), Lemma 4, the Lipschitz continuity of π, and (3). We now choose ǫ = λ ′ τ /3 (this means that C ǫ is now possibly quite large). Now choose η > 0 sufficiently small so that (C ǫ +C)η ≤ λ ′ τ /3. This means that d ′ (Ψ t x, s 0 • π(Ψ t x)) < τ for t > 0 sufficiently small which means thatΨ t x ∈ K as required.

Robustly Invariant sets in Jet Spaces
Here we apply the abstract results of the previous section to the setting of a C r+1 Anosov flow T t : M → M for some Riemannian manifold M. In other words the flow must satisfy the following condition.
The splitting is continuous and invariant with respect to T t . E f is one dimensional and coincides with the flow direction. In addition, there exists C, λ > 0 such that First we introduce jet spaces. Given two d s -dimensional hypersurfaces containing p ∈ M we say that they are equivalent if they have an r th -order contact at p. For each p the space of r + 1-jets is denoted J p and is defined to be the equivalence classes of all d s -dimensional hypersurfaces containing p. The hypersurfaces are said to represent the jet. The jet bundle is the set J := {(p, j) : p ∈ M, j ∈ J p }. The space J is a metric space with the metric d r (·, ·) defined as the inf of the C r+1 distance between hypersurfaces which represent the jets. This depends on the choice of coordinate charts and so we fix, once and for all, the choice of coordinate charts and use a partition of unity to insure that the metric varies smoothly from point to point on M. In fact, it is convenient to have co-ordinates charts in which all the relevant hypersufaces can be represented as graphs, then the C r distance would be simply the C r norm of a function. See [2, Section 3] for an explicit choice of such co-ordinates.
Note that every foliation of some region of M into C r+1 d s -dimensional hypersurfaces gives rise to a local section in J but not every section can be realised by a foliation of hypersurfaces. We have so defined our fibre bundle. There is an induced flow on the jet bundle Ψ t : J → J which we write as Ψ t : (p, j) → T −t (p), ψ t p j . If one considers a hypersurface through the point p ∈ M which is a representation of a jet j then ψ t p j is simply the jet which is represented by the image, under T −t , of that hypersurface. Notice that we are considering the flow in backwards time because we are ultimately interested in almost-stable objects.
By standard hyperbolic theory there exists a stable conefield (in tangent space) which we denote byK p at each p ∈ M and which is invariant under D p T −t0 for some t 0 > 0. We can naturally viewK p as a subset K p of J p . Hypersurfaces which represent the elements of K p are the hypersurfaces for which the tangent space at p belongs to K p . Let K = {(p, j) : p ∈ M, j ∈ K p } ⊂ J . It is a standard result of hyperbolic theory that there exists σ ∈ (0, 1) such that, perhaps increasing t 0 if required, 8 We now wish to apply the results of Section 2 to this setting and so we choose (J , d r ) as (E, d E ), K as E 0 , M as B, and d B as explained in Footnote 2. The existence of a Lipschitz section (s : M → J ) is easily achieved by taking the section which corresponds to the invariant stable foliation and then smoothing it. That the vector field associated to the Anosov flow has bounded C r+1 norm implies that (1) is satisfied. The estimate (7) implies that (2) is satisfied. Suppose that T t : M → M is another flow and that V andṼ are the corresponding vector fields of the two flows. Suppose furthermore that the C r+1 distance between the two vector fields is less than ζ for some ζ > 0. Then there exists t 1 > 0, C > 0 such that whereΨ t is the bundle flow corresponding toT t . This means that (3) is satisfied. All the above means that Theorem 1 implies the following.
Theorem 5. Suppose that T t : M → M is a C r+1 Anosov flow and J denotes the jet bundle associated to d s -dimensional submanifolds of M as described above. Then there exists J 0 ⊂ J and η > 0 such thatΨ t J 0 ⊂ J 0 for all t ≥ 0 and jet bundle flowsΨ t associated to a flow which is η-close to T t .

Errata Corrige to "Smooth Anosov Flows: Correlation Spectra and Stability"
In this section we show that the above theory provides a nice fix to the error in [2] explained in the introduction.
Before proceeding, let us remark that the problem in [2] can be eliminated in several ways. As already mentioned, a simple solution (used in [1,3]) is to keep the definition of leaves as in [2] and use the dynamics explicitly in the definition of the norm (by taking the new norm to be the sup of the old ones over some time interval). Unfortunately, this is not suited for the task at hand as we are interested also in perturbations of a given dynamics, hence we do not wish to have a norm tied too closely to the dynamic. One could probably fix the latter issue by taking the sup not only on time but also on a neighbourhood of time dependent dynamics. Yet, this would make the definition of the space rather cumbersome. Instead, we choose to redefine the set of admissible leaves by using the result in section 3. This has the merit of better elucidating the geometric properties of the dynamics and could be useful in related problems. Regrettably, in so doing we lose the compact embedding between the Banach spaces [2, Lemma 2.2], which previously held thanks to [4, Lemma 2.1]. The latter does not apply given the new definition of the leaves since it uses in a fundamental way the fact that, in charts, the tangent space of the leaves all belong to the same fixed cone. Nevertheless, it is easy to recover the needed compactness at the level of resolvent operators and hence obtain a correct proof of the results in [2].
Before starting let us note that [2, Condition 2] requires a strict contraction and expansion for all times for stable and unstable tangent vectors. This is used in the (incorrect) construction of the set of invariant leaves. In fact, what is actually needed is strict expansion in backward time for vectors belonging to the tangent space of the leaves. It is always possible to change the norm in such a way that [2, Condition 2] is satisfied but the strategy suggested to do so in [2, Footnote 1] is too naïve. At most, eliminating the absolute value in the exponent, one obtains a norm that contracts strictly along the stable direction. A norm with the claimed behaviour can be constructed as in [10], but then it has poor smoothness properties. For the reader's convenience and for completeness we show how to construct a smooth strictly-hyperbolic norm in Appendix A (at the price of not having the optimal contraction rate nor the orthogonality between invariant subspaces). Here, we content ourselves with a norm that has a strict backward expansion. This suffices for our present needs. Given the new construction of the set of admissible leaves provided shortly, the results of [2] holds under Condition 1, without worrying about adapted norms.

4.1.
A strictly expanding metric in tangent space. Let, as in [2], T η be the set of flowsT t defined by vector fields closer than η (in the C r+1 topology) to the vector field of T t . By Condition 1 we have that, for each λ ′ ∈ (0, λ), there exists η 0 > 0 such that there exists C 1 > 0 such that, for allT t ∈ T η0 , we have In analogy with (5) (and consistently with the aforementioned revision of [2, Footnote 1]) we can define, for some Lemma 6. There exists t 4 > 0 such that, for all p ∈ M, v ∈K p ,T t ∈ T η0 and t > 0, we have Proof. The proof is identical to the proof of Lemma 2 (apart from the inverse sense of the inequality) and holds provided C 1 e (λ ′ −λ ′′ )t4 > 1.
From now on we will use exclusively such a norm and we will suppress the prime to ease notation.

4.2.
The set of admissible leaves. We use Theorem 5 for r + 1-jets. Note that, by construction, the jets in J 0 at each p are represented by manifolds whose tangent space at p belongs toK p .
We call a d s -dimensional C r+1 manifold W pre-admissible if for each point p ∈ W there exists a neighbourhood of p such that the restriction of W to such a neighbourhood represents a r + 1-jet in J 0 . Let Σ 0 be the set of pre-admissible manifolds. Note that each manifold W ∈ Σ 0 has a natural Riemannian structure induced by the one of M. We can, and will, then talk about balls in W determined by such an induced metric. Given a fixed δ ∈ (0, 1) and R > 1, to be chosen shortly, we call the manifold W admissible if (1) W ∈ Σ 0 ; (2) W contains a ball of size δ; (3) W has diameter smaller than Rδ. Let Σ be the collection of admissible manifolds (leaves). Here we come to the basic result of this errata: to restrict the set of allowed manifolds so that it has the wanted invariance also for small times and so that the property persists under perturbations.
Lemma 7. There exists η 0 > 0 such that, for eachT t ∈ T η0 , leaf W ∈ Σ and t ∈ R + , there exist leaves W 1 , . . . , W ℓ ∈ Σ, whose number ℓ is bounded by a constant depending only on t, such that There exists a constant C (independent of W and t) such that each point ofT −t W + is contained in at most C sets W j . (4) There exist functions ρ 1 , . . . , ρ ℓ of class C r+1 , ρ j compactly supported on W j , such that ρ j = 1 onT −t (W ), and |ρ j | C r+1 ≤ C.
Proof. Theorem 5 immediately implies that the DT −t Σ 0 ⊂ Σ 0 . If W ∈ Σ, then Lemma 6 implies that DT −t W satisfies the second requirement of an admissible leaf for all t > 0. Unfortunately, DT −t W it may grow too much and fail to satisfy the third condition. In such a case note that the manifolds W ′ ∈ T −t Σ have a uniformly bounded curvature and, by choosing δ small enough, we can assume that each one of them belongs to some chart of M. Then there exists K > 0 such that the distance between two point on W ′ are less than K times, and more than K −1 times, the Euclidean distance in the chart. If p is a point of the manifold at a distance larger than δ from the boundary, we can consider, in the chart, an (Euclidean)-ball of centre p and radius Kδ. Let us call W ′′ the intersections of the manifold with such a ball. By construction W ′′ will contain a ball (in the induced Reimannian metric) of size at least δ (hence satisfying property (2)). On the other hand, the diameter of W ′′ will be less than K 2 δ, hence satisfying property (3) [4], as far as we see [4, Lemma 2.1] could be false in the present context. Indeed, the proof in [4] uses in a crucial way that in each chart the cone field is constant, a property that we no longer have. To overcome this obstacle we bypass [2, Lemma 2.2] and provide a direct proof of [2,Lemma 4.4] following the strategy in [9]. This provides a complete proof of the results in [2].
Lemma 8. For each p ∈ N, q ∈ R + , q + p < r, and z ∈ C, ℜ(z) = a > 0 the operator R(z) : B p,q → B p,q has spectral radius bounded by a −1 and essential spectral radius bounded by (a +pλ) −1 .
Proof. Since the conclusions of [2, Lemmata 4.1, 4.3] hold, the estimate of the spectral radius is as before. It remains to prove the bound on the essential spectral radius. First of all notice that there exists K > 0 such that L t ∈ L(B p,q ,B p,q ) and L t ∈ L(B p−1,q+1 , B p−1,q+1 ) for all t ≥ K. 9 From this it follows that the operators are compact, as operators in L(B p,q , B p−1,q+1 ). Indeed, the incorrect proof of the compactness of R(z) in [2] holds correct for the operator in square brackets, since no small times are involved, yielding compactness as an operator in L(B p,q ,B p−1,q+1 ). The result then follows by the above continuity properties of L K . Thus, setting we have R(z) m = R K,m + Q K,m , and [2, Lemma 4.1] implies We can then conclude by the usual Hennion's argument [6] based on Nussbaum's formula [11]. Let us recall the argument. Let B = {h ∈ B p,q : h p,q ≤ 1} and B m = R K,m (z)B. By the above discussion B m is compact in B p−1,q+1 . Thus, Choosing ǫ = a n (a + λp) −n+1 and m = δn, for δ small enough, we conclude that, for each λ ′′ ∈ (0, λ), for each n ∈ N the set R(z) n B can be covered by a finite number of · p,q -balls of radius C(a +pλ ′′ ) −n , which implies that the essential spectral radius of R(z) cannot exceed (a +pλ) −1 .

Appendix A. Strict hyperbolicity
In this appendix we show that if a flow satisfies Condition 1, then there exists an equivalent smooth metric such that [2, Condition 2] is satisfied. Note that the content of this appendix is not required for any of the results of the previous sections.
Lemma 9. For each C r+1 Anosov flow there exist a C r Riemannian metric · 1 , uniformly equivalent to the original Riemannian metric, and σ > 0 such that where C u (x), C s (x) are invariant unstable and stable conefields, respectively.
Proof. By Theorem 5 (applied both to T t and T −t ) we have the existence of invariant conesC u ,C s . Then, by assumption, there exists λ > 0, C > 0 such that The construction of the norm is based on the parameter L > 0, chosen such that Ce 2λL > 1. We define the new metric We consider the case v ∈ C u := DT LC u and will prove the first inequality of the lemma. By the semigroup property of the flow it suffices to prove the statement for t ∈ [0, L].
where, in the last line, we used (8) and the definition of C u . Note that, for each L > 0 there exists C L > 0 such that Combining this with the previous estimate we have shown that, for all t ∈ [0, L], Since C 2 e 4λL − 1 > 0 (because L > 0 was chosen appropriately) this means that there exists σ > 0 such that DT t v 1 ≥ e σt v 1 as required.
For the other inequality we can argue in complete analogy with the above computation but with time reversed.
What we have done is to find a norm for which (8) holds with C = 1 (with a different λ > 0). This is very similar to what is proven in [10] for the case of Anosov diffeomorphisms but here there are two (closely related) differences: 1) the metric ·, · 1 is C r rather than just Hölder as in [10]; 2) contrary to [10] the new distributions are not orthogonal in the new norms. The latter is annoying but inevitable if one wants a smooth norm.
Note that, in the above proof, σ cannot be taken arbitrarily close to λ, contrary to [10]. Indeed (10) suggests that typically σ will be much smaller than λ. This is the price of the naïvity of our construction and the requirement that the metric be smooth.