doi: 10.1017/S0143385707000272 Printed in the United Kingdom
Adapted metrics for dominated splittings
NIKOLAZ GOURMELON
I.M.B., UMR 5584 du CNRS, B.P. 47 870, 21078 Dijon Cedex, France (e-mail: [email protected])
(Received14January2005and accepted in revised form23September2005)
Abstract. A Riemannian metric isadaptedto a hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction of the unstable/stable directions is seen after only one iteration. Adominated splittingis a notion of weak hyperbolicity where the tangent bundle of the manifold splits in invariant subbundles such that the vector expansion on one bundle is uniformly smaller than that on the next bundle. The existence of anadapted metricfor a dominated splitting has been considered by Hirsch, Pugh and Shub (M. Hisch, C. Pugh and M. Shub.Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977). This paper gives a complete answer to this problem, building adapted metrics for dominated splittings and partially hyperbolic splittings in arbitrarily many subbundles of arbitrary dimensions. These results stand for diffeomorphisms and for flows.
1. Introduction
The best known and simplest examples of chaotic dynamical systems are uniformly hyperbolic systems, like Anosov diffeomorphisms. A diffeomorphism f on a compact Riemannian manifoldMis said to be anAnosov diffeomorphismif there exists a splitting of the tangent bundleT Minto two supplementary,d f-invariant subbundles, called thestable and theunstablebundles that are uniformly contracted and expanded, respectively, by an iterate of f. If the hyperbolic systems are now well understood, many dynamical systems are (robustly) not hyperbolic, so that several authors have tried to weaken the notion of hyperbolicity, in order to recover some of its properties on a larger class of systems.
Working on the stability conjecture†, Liao [Lia80], Ma˜n´e [Man82] and Pliss [Pli72] were led to the following general notion: adominated splittingfor f is a splitting ofT M into two supplementary invariant subbundles such that there exists an iterate ofd f that uniformly contracts more (or expands less) the first subbundle than the second one. This notion is a key tool for understanding non-hyperbolic systems.
• In dimension two, Pujals and Sambarino [PS00] proved that a diffeomorphism with a dominated splitting may beC1-approached by hyperbolic diffeomorphisms, and diffeomorphisms without dominated splitting may be approached by diffeomorphisms exhibiting a homoclinic tangency: as a consequence, any diffeomorphism of a compact surface can beC1-approximated either by hyperbolic (Axiom A) diffeomorphisms, or by diffeomorphisms that exhibit a homoclinic tangency (this was conjectured by Palis [PS70]).
• In any dimension, Bonattiet alshowed in [BDP00] that a robustly transitive generic diffeomorphism in Diff1(M)admits a non-trivial dominated splitting defined on the wholeM.
As recalled above, for a hyperbolic set K of a diffeomorphism f, the vectors in the stable and unstable bundles are uniformly contracted and expanded, respectively, by the derivatived fn, for somen>0. The hyperbolicity ofK does not depend on the metric on the manifold, but the smallest timenwhere the contraction/expansion phenomena are seen depends on the metric; a Riemannian metric is said to beadaptedto the hyperbolic setK if one can taken=1. Applying Holmes’ theorem (see [HPS77, p. 15]), we obtain that any hyperbolic set admits an adapted Riemannian metric. We will adapt this theorem to the case of dominated behaviours, to show Lemma 4.1. It was asked in [HPS77, p. 5] if there existed anadapted metric for a dominated splitting, that is a metric such thatd f uniformly contracts (or respectively expands) the first subbundle more (or respectively less) than the second one, at the first iteration.
The aim of this paper is to give a complete positive answer to this question, proving that such an adapted metric exists for any dominated splitting.
THEOREM1. Let f be a diffeomorphism of a Riemannian manifold M, and K a compact invariant subset of M, such that the restriction of f to K admits a dominated splitting T M|K=E1⊕E2⊕ · · · ⊕Ed, where the vectors in Ei are uniformly less expanded than
those in Ei+1by d fnfor some n>0. Then there exists a Riemannian metrick · kon M (necessarily equivalent to the first metric) and adapted to the dominated splitting: there exists a constant0< µ <1such that for any x∈K , any i∈ {1, . . . ,d−1}, and any unit vectors u∈Eix,v∈Ei+1x , one haskd f(u)k< µ.kd f(v)k.
This result was already known by Hisch et al[HPS77] for a dominated splitting in two bundles,T M|K=E1⊕E2, such that dim(E1)=1 or dim(E2)=1. In addition, they
showed that any absolutely normally hyperbolic system admits an adapted metric, but it was not known whether it was true for a relatively normally hyperbolic system, which is, with our definitions, a partially hyperbolic system. We answer by showing (see Theorem 3)
that an adapted metric exists for any partially hyperbolic splitting, that is a metric adapted to the corresponding dominated splitting, and such that the stable/unstable bundles are uniformly expanded/contracted at the first iterate. Finally, we show in §5 how to transpose these results from diffeomorphisms to flows.
In order to present more clearly the idea of the proof, we will first focus, in §3, on dominated splittings into two subbundles, over an invariant compact set of a diffeomorphism. Then, we will show that there exists an adapted Finsler for any dominated splitting intod bundles for a Banach bundle automorphism (see Theorem 2).
2. Definition and notations
For a morphismAof normed vector spaces, define thenormand theminimum normofA: kAk = sup
kuk=1
kA(u)k, m(A)= inf
kuk=1kA(u)k.
When Ais invertible, m(A)= kA−1k−1. For a Banach bundle E, we denote by Ex the
fibre ofEabove a pointxof the base. IfAis an automorphism of a Banach bundleEwith compact baseK, then, for any pointxof K, we denote byAx the restriction ofAto the fibreEx. We refer the reader to [HPS77] for definitions.
We say that a sequence of functionsgn(x): K→Rconverges exponentially to zeroif
there exists positive constantsCandµ <1 such that, for allxandn, |gn(x)| ≤Cµn.
Given an automorphismAof a Banach bundleEwith compact baseK
E A //
π
E
π
K f //K
and a positive continuous functionr: K →R, we denote byRn(x)the product
Rn(x)= n−1
Y
i=0
r[fi(x)] =r(x)r[f(x)] · · ·r[fn−1(x)].
Definition 2.1. A positive continuous functionr: K→RdominatesA, if the sequence of
ratiosx→ kAn
xk/Rn(x)converges exponentially to zero asn→ ∞, whereAnx=An|Ex.
In this case we writed f|E≺r. Symmetrically, we say thatrisdominatedbyAand we
writer≺Aif and only if the ratioRn/m(An)goes exponentially to zero asn→ ∞. Note
thatr≺Ais equivalent toA−1≺1/r.
Definition 2.2. LetEbe a Banach bundle over a compact baseK, andE=E1⊕ · · · ⊕Ed be an invariant splitting for an automorphism A, where the Ei are vector subbundles with constant dimension. Then we say that it is adominated splittingif, for each integer 0<i<d, the ratiokAn
|Eik/m(A
n
We have written kAn
|Eik/m(An|Ei+1)for the function x7→ kAn|Ei x
k/m(An
|Exi+1). In this case, we say thatA|Ei isdominatedbyA|Ei+1, and we writeA|Ei ≺A|Ei+1. We recall that the subbundlesEiare necessarily continuous (see [BDV04, Appendix B] for a proof). Remark 2.3. Since the bundles and automorphisms are continuous, and the base K is compact, the definitions of domination and dominated splitting are independent of the Finsler. Thus we will be allowed to change to equivalent metrics; in finite dimension, all Finslers can be replaced by smooth Riemannian metrics.
A Finsler k · k∗ isadaptedto the dominated splitting if and only if, for alli<d, we
have
kA|Eik∗
m∗(A
|Ei+1) <1
where k · k∗ andm∗ are the norm and the minimum norm, with respect to the Finsler
k · k∗. Equivalently, by compactness of the base, there exists a real number 0<C<1
such that, for any x∈K, for any non-zero unit vectors u∈Eix, v∈Exi+1, we have kA(u)k∗<CkA(v)k∗.
3. Two-bundle splittings
Let M be a compact smooth manifold endowed with a Riemannian metrick · k, let f be a diffeomorphism of M and let K be an invariant compact set in M. We will show the following theorem.
THEOREM3.1. If TKM=E⊕F is a dominated splitting for the diffeomorphism f on
the compact K , then there exists a smooth Riemannian metric on M that is adapted to that dominated splitting.
Proof. The proof consists in first building aseparatorfor the dominated splitting, that is, a positive functionr: K→Rsuch that we haved f|E≺r≺d f|F. Then, by a dominated
version of Holmes’ theorem, we will build two metricsk · kE andk · kFon the bundlesE
andF, such that, for anyx∈K, for any unit vectorsu∈Ex andv∈Fx, we have
kd f(u)kE<r(x) <kd f(v)kF.
These metrics will induce, up to perturbation, an adapted Riemannian metric onM. LEMMA3.1. A two-bundle dominated splitting has a separator.
Proof. In the following, we fix a dominated splitting TKM=E⊕F for the
diffeomorphism f. For simplicity, calld f|E=Aandd f|F=B. By hypothesis, the ratio
kAnk/m(Bn)tends exponentially to zero. In particular, for N large enough, the function
x7→ kANxk/m(BN
x )is smaller than 1/2. Therefore, fora>1>bclose enough to one, we
have for allxinK:
akAxNk1/N<bm(BN
x)1/N.
Hence, Lemma 3.1 comes from the following claim. ✷
CLAIM1. Any continuous function r: K→Rsuch that a.kAxNk1/N≤r(x)≤b.m(BN
x )1/N
Proof. For any integern>N, and each 0≤k≤N−1, we can write the iterateAn xas the
composition
Al
fk+m N(x)◦A
m N fk(x)◦A
k x
for some integers 0≤l≤N−1 and m≥0. More precisely, take the integer part of (n−k)/N for m, andl=n−n M−k. Denote byc the upper bound of the norms of theith forward or backward iterates of A, fori≤N,
c= sup
|i|≤N,y∈K
kAiyk. It is finite, asKis compact. We then have
kAnxk ≤ kAl
fk+nm(x)k m−1
Y
i=0
kAN
fk+i N(x)k
kAkxk kAnxk ≤c2 Y
j∈Jk kAN
fj(x)k
(1k)
where Jk is the set of integers {k+i N,i=0, . . . ,m−1}, that is, the set of integers
of the form k+i N and comprised between 0 and n−N. Obviously, the sets Jk for
k=0, . . . ,N−1 are pairwise disjoint and their union is the interval{0, . . . ,n−N}. Hence, taking the product of inequalities (1k), fork=0, . . . ,N−1 we obtain
kAnxkN≤c2N Y
j∈{0,...,n−N}
kAN
fj(x)k. SincekAN
fj(x)k
−1≤ kA−N
fj+N(x)k ≤c, we get kAnxkN≤c2NcN Y
j∈{0,...,n}
kAN
fj(x)k. Thus, asakAN
fj(x)k1/N≤r[fj(x)], we get that, for anyx∈K, kAnxk ≤c3a−nRn(x),
which proves thatA≺r. Note that(1/b).kB−Nx k1/N≤1/r[FN(x)], for allx. Thus, we have in the same wayB−1≺1/r and thenr≺B. This ends the proof of the claim, and
that of Lemma 3.1. ✷
We now show the following lemma (which can actually be seen as a particular case of Lemma 4.1 stated below).
LEMMA3.2. Let r: K→Rbe a positive function that separates the continuous splitting E⊕F , that is, d f|E≺r≺d f|F. Then there exists a Riemannian metrick · k∗on M that
isadaptedto the domination; namely, for all x∈K , for all unit vectors u∈Ex,v∈Fxwe
have
kd f(u)k∗<r(x) <kd f(v)k∗.
Proof. We define on Ea metrick · kEby
kuk2E=
∞
X
n=0
for anyu∈Ex, whereRn(x)=r(x)· · ·r[fn−1(x)]as above. By domination, this is a sum
of a normally convergent series of continuous functions; thereforek · kE is well defined
and continuous. As a sum of quadratic forms,k · k2E is a quadratic form, and thusk · kEis
a Hilbertian metric (it arises from an inner product). Moreover, we have kd f(u)k2E=
∞
X
n=0
kd fn+1(u)k2 [Rn(x)]2
=
∞
X
n=1
kd fn(u)k2 [Rn−1(x)]2
=r(x)2
∞
X
n=1
kd fn(u)k2 [Rn(x)]2
since Rn−1(x)=Rn(x)/r(x). We obtainkd f(u)k2E=r(x)2[kuk2E− kuk2]wherekuk2
is the first term of the series defining kuk2E. Hence, for any non-zero u, kd f(u)kE<
r(x)kukE. Up to change of f into f−1andrinto 1/r, we find the same way a Hilbertian
metric k · kF on F such that, for all non-zerov in F,r(x)kvkF<kd f(v)kF. Consider
now the Hilbertian metrick · k∗onTKM that extendsk · kE andk · kF and that makesE
andForthogonal. It is continuous, sincek · kEandk · kFare continuous.
The inequality kd f(u)k∗<r(x) <kd f(v)k∗ holds for all unit vectorsu∈E, v∈F
above each pointxof the baseK. We extend the metrick · k∗to the wholeM, and smooth
it into a Riemannian metric by a small perturbation, so that, by compactness of K, the
inequality is preserved. ✷
This together with the existence of a separator (Lemma 3.1) ends the proof of
Theorem 3.1. ✷
4. Multiple bundles splittings
We will show the most general result of our paper in this section.
THEOREM2. Let E be a finite-dimensional Banach bundle on a compact base, and let
Abe an automorphism of E . If E=E1⊕ · · · ⊕Ed is a dominated splitting forA, then
there is a Finslerk · k∗on E adapted to the domination, that is, for each i=1. . .d−1,
for any x∈K , we have
kA|Ei xk∗<
m∗(A
|Ei+1x ).
Furthermore, if the original metric on E is Hilbertian, then the adapted metric can be chosen to be also Hilbertian.
LetFbe a Banach bundle with compact baseK, andBbe an automorphism
F B //
π
F
π
K f //K
Then we have this dominated version of Holmes’ theorem (see [HPS77, p. 15]).
LEMMA4.1. Let r,s: K→R be two positive continuous functions such that the domination r≺B≺s and the inequality r<s hold on K . Then there is a Finslerk · k∗on
F that is adapted to the domination, namely, for any x∈K , for any u∈Fx\ {0}, we have
Proof. For alluinF, we define
kuk2∗=
∞
X
n=1
Rn2[f−n(x)]kB−n(u)k2+
∞
X
n=0
kBn(u)k2 [Sn(x)]2
wherek · kis the original metric onF, and whereRn[f−n(y)] =r(f−n(y))· · ·r(f−1(y)),
Sn(y)=s(y)s(f(y))· · ·s(fn−1(y))as before. Still by domination, the series normally
converges to a continuous function. Thusk · k∗is a well-defined Finsler. We have
kB(u)k2∗=
∞
X
n=1
Rn2[f−n+1(x)]kB−n+1(u)k2+
∞
X
n=0
kBn+1(u)k2 [Sn(f(x))]2
=
∞
X
n=0
Rn+12 [f−n(x)]kB−n(u)k2+
∞
X
n=1
kBn(u)k2
[Sn−1(f(x))]2
.
As we haveRn+1[f−n(x)] =Rn[f−n(x)].r(x)andSn−1[f(x)] =Sn(x)/s(x), we get
kB(u)k2∗= [r(x)]2
∞
X
n=0
Rn2[f−n(x)]kB−n(u)k2+ [s(x)]2
∞
X
n=1
kBn(u)k2
[Sn(x)]2
.
On the other hand, we haveR0(x)kB−0(v)k = kvk = kB0(v)k/S0(x)sinceR0andS0are
empty products equal to one. So, from the definition ofkuk2∗, we have
kuk2∗=
∞
X
n=0
Rn2[f−n(x)]kB−n(u)k2+
∞
X
n=1
kBn(u)k2
[Sn(x)]2
.
Finally, sincer<s, we obtain, for any non-zero vectoru, the inequality [r(x)]2kuk2∗<kB(u)k2∗<[s(x)]2kuk2∗
the square root of which concludes the proof. ✷
Remark 4.2. Having chosen this quadratic construction, ifk · kis a Hilbertian metric, then k · k∗is still a Hilbertian metric.
Proof of Theorem 2. By definition, the ratioskAn
|Eik/m(An|Ei+1)converge exponentially to zero, for eachi, asngoes to infinity. Thus we can find an integerN such that, for eachi, the ratiokAN
|Eik/m(A|ENi+1)is smaller than 1/4. The proof of Lemma 3.1 still works when T M|K=E⊕F is replaced by the Banach bundleEi ⊕Ei+1, and whend f|K is replaced
by the automorphismA|Ei⊕Ei+1.
Then choose a family(ri)0<i<d of continuous functions such that 21/NkA|ENik1/N< ri <2−1/Nm(AN
|Ei+1)
1/N. We then haver
1(x) <· · ·<rd−1(x), for all x∈K;
further-more, by Claim 1, we have
E1≺r1≺E2≺ · · · ≺rd−1≺Ed.
a Finslerk · ki on eachEi that is adapted to the dominationri≺Ei+1≺ri+1. Define the
new metric
kuk∗=
s X
i=1,...,d
kpi(u)k2i,
for allu∈E, where pi is the projection onEi alongE1⊕ · · · ⊕Ei−1⊕Ei+1· · · ⊕Ed.
It is a Finsler that is clearly adapted to the dominated splitting: for any unit vectorsu∈Eix, v∈Exi+1we havekuk∗= kuki<ri(x) <kvki+1= kvk∗.
Remark 4.3. Obviously by the previous remark, if the original metrick · kwas a Hilbertian metric, then the metricsk · ki are so, and the metrick · k∗we have built is still a Hilbertian
metric.
After smoothing the adapted Hilbertian metric, we obtain the following, which is a reformulation of Theorem 1.
COROLLARY4.4. Let M be a Riemannian manifold, K a compact invariant set for a diffeomorphism f , and TKM=E1⊕ · · · ⊕Eda dominated splitting for f above K . Then
there exists a smooth Riemannian metric on M that is adapted to it.
The existence of an adapted metric was shown forabsolute-and notrelative-normally hyperbolic systems (see [HPS77] for proofs and definitions). The bases of all Banach bundles are still compact. A dominated splittingE=E1⊕ · · · ⊕Edfor an automorphism
A is partially hyperbolic if and only if, for some 1≤k<k+1<l≤d, the bundles Es=E1⊕ · · · ⊕Ek andEu=El⊕ · · · ⊕Ed are respectively stable and unstable, that
iskAn
|Esk andkA−n|Eukconverge exponentially to zero as n goes to infinity. We say that a metric k · k∗ isadapted to such a partially hyperbolic splitting if it is adapted to the
dominated splitting, and ifkA|Esk<1 andm(A|Eu) >1.
THEOREM3. A partially hyperbolic splitting has an adapted metric.
Proof. We show it in the three-bundle case (it is the same idea for the general case). Consider a partially hyperbolic splitting E=Es⊕Ec⊕Eu with compact base for an automorphismA. ThenkAn
|EskandkA−n|Euktend exponentially to zero asngoes to infinity. From the construction we gave in the proof of Lemma 3.1, we can find two functions 0<r<1<ssuch that
A|Es ≺r≺A|Ec≺s≺A|Eu.
With respect to this domination, the metric k · k∗ produced in the proof of Theorem 2
is adapted to the dominated splitting and satisfies kA|Esk<1<m(A|Eu). Hence, it is
adapted to the partially hyperbolic splitting. ✷
5. Dominated splittings for flows
In this section, we briefly show that the same results apply for flows. In the following,φ is a flow on a compact subset K of a Riemannian manifold M. A Finslerk · k∗ onM is
adaptedto a dominated splittingTKM=E1⊕ · · · ⊕Edforφ, if and only if, for any point
wheredφt is the derivative of the time-t map ofφ. The existence of adapted metrics for flows is not a straightforward consequence of our results on diffeomorphisms. At best, applying the former results would provide, for eachǫ >0, a metrick · k∗ such that the
inequality above holds for all t> ǫ. To get adapted metrics, we have to transpose the notion of separator to the flow case.
Let E be a subbundle of TKM, invariant byφ. Fix two strictly positive, continuous
functionsr,s: K→R. Then, for anyx∈K, for allt∈R, define Rt(x)=exp
Z t
0
ln[r(φu(x))].du
,
St(x)=exp
Z t
0
ln[s(φu(x))].du
.
For any fixedx, the functionst7→Rt(x)andt7→St(x)areC1, and for all real numbers
t,k,
Rt+k(x)=Rt(x).Rk[φt(x)], (1)
St+k(x)=Rt(x).Sk[φt(x)]. (2)
Assume that r<s, and that we have the domination relation r≺dEφ≺s, that is, for
allx∈K, for any vectorv∈Ex, the quantitieskdφt(v)k/St(x)andkdφt(v)k/Rt(x)go
exponentially to zero, respectively, ast goes to+∞, and ast goes to−∞. Then, define the Finslerk · k∗onEby
kvk2∗= Z 0
−∞
kdφt(v)k2 Rt(x)2
.dt+ Z ∞
0
kdφt(v)k2 St(x)2
.dt
for anyx∈K, for allv∈Ex.
CLAIM2. For any non-zero vectorv∈E above any point x∈K , the metrick · k∗satisfies
for all k>0, Rk(x).kvk∗<kdφk(v)k∗<Sk(x).kvk∗.
That is, the metrick · k∗isadaptedto the dominationr≺dEφ≺s.
Remark 5.1. This is merely Lemma 4.1 for flows. Proof. After a change of variable, we get
kdφk(v)k2∗= Z k
−∞
kdφt(v)k2 R2t−k◦φk(x).dt+
Z +∞
k
kdφt(v)k2 St−k2 ◦φk(x).dt
=Rk2(x) Z k
−∞
kdφt(v)k2 Rt2(x) .dt+S
2 k(x)
Z +∞
k
kdφt(v)k2 St2(x) .dt, by formulae (1) and (2). Letθ (k)be the quotientSk2(x)/Rk2(x), and define the function
f: k7→ kdφ
k(v)k2 ∗
Rk2(x) = Z k
−∞
kdφt(v)k2
R2t(x) .dt+θ (k) Z +∞
k
kdφt(v)k2 St2(x) .dt.
f′(k)=θ′(k) Z +∞
k
kdφt(v)k2 St2(x) .dt.
Hence f′ is strictly positive, and f is strictly increasing. As we have f(0)= kvk2∗/R02(x)= kvk2∗, the inequality f(k) > f(0)leads to Rk(x)2.kvk2∗<kdφk(v)k2∗, for
allk>0. The inequalitykdφk(v)k∗2<Sk2(x).kvk2∗comes the same way, considering this time the functiong: k7→ kdφk(v)k2∗/Sk2(x)
. This concludes the proof of the claim. ✷
On the other hand, any dominated splitting for a flow has a separator. Let TKM=
E⊕Fbe a dominated splitting. For simplicity,8will denote the restriction ofdφto the bundle E. For instance, we will writek8txkfor the maximum norm of the restriction of dφtoEx. More precisely, we assert the following claim.
CLAIM3. The function r= x7→a.k8Txk1/T
is a separator E≺r≺F , for some a>1, and some T large enough.
Remark 5.2. This is Lemma 3.1 for flows; the proof is comparable step by step to that of Claim 1.
Proof. Fix a realt>T. Letmbe the largest integer such thatT +mT ≤t. For any real 0≤κ≤T, we decompose8tto obtain, for allx∈K,
k8txk ≤ k8λ
φκ+mT(x)k · k8Tφκ+(m−1)T(x)k · · · k8φTκ+T(x)k · k8φTκ(x)k · k8κxk,
whereλ >0 satisfiest=κ+mT+λ. Denote bycthe upper boundc=sup|τ|≤2T,y∈K k8τyk, and take the logarithm of the inequality,
lnk8txk ≤2 ln(c)+lnk8T
φκ+(m−1)T(x)k + · · · +lnk8
T
φκ(x)k.
This stands for all 0≤κ≤T. Therefore, we have Z T
0
lnk8txk.dκ ≤ Z T
0
2 ln(c).dκ+ Z T
0
lnk8T
φκ+(m−1)T(x)k.dκ + · · · +
Z T
0
lnk8Tφκ(x)k.dκ,
T lnk8txk ≤2T ln(c)+ Z mT
0
lnk8Tφu(x)k.du,
T lnk8txk ≤4T ln(c)+ Z t
0
lnk8Tφu(x)k.du,
since 0≤t−mT ≤2T, and−lnk8Tyk ≤lnk8−Ty k ≤ln(c). Then, dividing by T, we obtain
lnk8txk ≤ln(c4)+ Z t
0
ln[a−1.r(φu(x))].du.
Writing the exponential form, we get k8txk ≤c4a−tRt(x), which means that8=dEφ
≺r. We are left to check thatr≺dFφfor someT >0 big enough, and somea>1. This
is done the same way as in Lemma 3.1. ✷
Clearly, referring the reader to the proof of Theorem 2, flow versions of Lemmas 3.1 and 4.1 are all the ingredients we need, to transpose the results of §4 to flows.
Acknowledgements. I would like to thank Christian Bonatti and Sylvain Crovisier for second reading and corrections. Special thanks also go to the referee for his great help in writing this paper.
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