Admissibility and nonuniformly hyperbolic sets

Admissibility and nonuniformly hyperbolic sets

Luis Barreira

, Davor Dragiˇcevi´c

and Claudia Valls

1Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal

2School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia

Received 19 November 2015, appeared 7 March 2016 Communicated by Michal Feˇckan

Abstract. We obtain a characterization of two classes of dynamics with nonuniformly hyperbolic behavior in terms of an admissibility property. Namely, we consider expo- nential dichotomies with respect to a sequence of norms and nonuniformly hyperbolic sets. We note that the approach to establishing exponential bounds along the stable and the unstable directions differs from the standard technique of substituting test se- quences. Moreover, we obtain the bounds in a single step.

Keywords: exponential dichotomies, nonuniformly hyperbolic sets.

2010 Mathematics Subject Classification: 37D99.

1 Introduction

Our main objective is to obtain a characterization of two classes of dynamics with nonuni- formly hyperbolic behavior in terms of an admissibility property. Namely, we consider the class of exponential dichotomies with respect to a sequence of norms and the class of nonuni- formly hyperbolic sets.

In the first part of the paper we consider a nonautonomous dynamics with discrete time obtained from a sequence of linear operators on a Banach space and we characterize the notion of an exponential dichotomy with respect to a sequence of norms. The principal motivation for considering this notion is that includes both the notions of a uniform and of a nonuniform exponential dichotomy as special cases. We refer the reader to the books [3,6,7,12] for details and further references on the uniform theory. On the other hand, the requirement of unifor- mity for the asymptotic behavior is often too stringent for the dynamics and it turns out that the notion of a nonuniform exponential dichotomy is much more typical. We refer the reader to [2] for an account of a substantial part of the theory. Most of the work in the literature related to admissibility has been devoted to the study of uniform exponential dichotomies.

For some of the most relevant early contributions in the area we refer to the books by Massera and Schäffer [10] and by Dalec⁰ki˘ı and Kre˘ın [4]. We also refer to [9] for some early results in infinite-dimensional spaces. For a detailed list of references, we refer the reader to [3] and for more recent work to Huy [8].

BCorresponding author. Email: barreira@math.tecnico.ulisboa.pt

We emphasize that we consider the general case of a noninvertible dynamics which means that we assume only the invertibility along the unstable direction. Moreover, we characterize exponential dichotomies with respect to a sequence of norms in terms of the admissibility of a large family of Banach spaces (the particular case ofl^pspaces was considered in [1]). We note that the approach to establishing exponential bounds along the stable and the unstable direc- tions differs from the standard technique of substituting test sequences (see for example [6,8]).

Moreover, in contrast to the existing approaches, we are able to obtain bounds along the stable and unstable directions in a single step.

In the second part of the paper we obtain an analogous characterization of nonuniformly hyperbolic sets. The notion of a nonuniformly hyperbolic set arises naturally in the context of smooth ergodic theory. Indeed, if f is a C¹ diffeomorphism of a finite-dimensional compact manifold preserving a finite measure µwith nonzero Lyapunov exponents, then there exists a nonuniformly hyperbolic set of full µ-measure. We refer the reader to [2] for details. Our work is close in spirit to that of Mather [11], who obtained a similar characterization of uni- formly hyperbolic sets, as well as that of Dragiˇcevi´c and Slijepˇcevi´c [5], where the problem of extending Mather’s result to nonuniformly hyperbolic dynamics was first considered. How- ever, there are substantial differences between our approach and that in [5], which provides a characterization of ergodic invariant measures with nonzero Lyapunov exponents and not of nonuniformly hyperbolic sets.

2 Preliminaries

In this section we introduce a few basic notions. LetS be the set of all sequencess= (s_n)_n∈Z of real numbers. We say that a linear subspaceB⊂ S is anormed sequence spaceif there exists a normk·k_B: B→_R⁺₀ such that ifs⁰ ∈ Band|s_n| ≤ |s⁰_n|forn∈_{Z, then}s∈ Bandksk_B ≤ ks⁰k_B. If in addition(B,k·k_B)is complete, we say thatBis aBanach sequence space.

LetBbe a Banach sequence space. We say that Bisadmissible if:

1. χ_{n} ∈ Bandkχ_{n}k_B >0 forn∈_{Z, where}χ_Adenotes the characteristic function of the setA⊂_Z;

2. for eachs = (_sn)_n∈Z ∈ Bandm∈ Z, the sequences^m = (_s^m_n)_{n∈Z defined bys}^m_n = _sn+m belongs toBand there exists N>0 such thatks^mk_B ≤ Nksk_B fors∈Bandm∈_Z.

We present some examples of Banach sequence spaces.

Example 2.1. The set l^∞ = {s ∈ S : sup_n∈Z|s_n| < +_∞} is a Banach sequence space when equipped with the normk_sk=sup_n∈Z|s_n|.

Example 2.2. For each p∈[1,∞), the setl^p={_s∈ S :∑n∈_Z|s_n|^p<+_∞}is a Banach sequence space when equipped with the normksk= (_∑n∈Z|sn|^p)^1/p.

Example 2.3. Letφ: (0,+_∞)→(0,+_∞]be a nondecreasing nonconstant left-continuous func- tion. We setψ(t) =Rt

0φ(s)ds fort ≥ 0. Moreover, for eachs ∈ S, let M_φ(s) =_∑n∈Zψ(|s_n|). Then

B=s∈ S : Mφ(cs)< +_∞for somec>0 is a Banach sequence space when equipped with the norm

k_sk=inf

c>0 :M_φ(s/c)≤1 .

We need the following auxiliary results.

Proposition 2.4. Let B be an admissible Banach sequence space.

1. If s¹ = (s¹_n)_n∈Z ands² = (s²_n)_n∈Z are sequences in S and s¹_n = s²_n for all but finitely many n∈ _{Z, then}s¹∈ B if and only ifs²∈ B.

2. Ifsⁿ→_sin B when n→_∞, then sⁿ_m →s_m when n→_∞, for m∈_Z. 3. For eachs∈B andλ∈(0, 1), the sequencess¹ands² defined by

s¹_n=

∑

m≥0

λ^ms_n−m and s²_n =

∑

m≥1

λ^ms_n+m

are in B, and

ks¹k_B ≤ ^N

1−λksk_B and ks²k_B ≤ ^Nλ

1−λksk_B. (2.1) Proof. 1. Assume that s¹ ∈ B and let I ⊂ _Zbe the finite set of all integers n ∈ _Zsuch that s¹_n 6= s²_n. We define v = (v_n)_n∈Z by v_n = 0 ifn ∈/ I andv_n = s²_n−s¹_n if n ∈ I. Since B is an admissible Banach sequence space, we havev∈Band thuss²=_s¹+_v∈ B.

2. We have

|sⁿ_m−s_m|χ_{m}(k)≤ |sⁿ_k −s_k|

fork∈ _Zandn∈N. By the definition of a normed sequence space, we obtain

|sⁿ_m−s_m| ≤ ^N

kχ_{0}k_Bksⁿ−sk_B forn∈_Zand the conclusion follows.

3. We define a sequence v = (v_n)_n∈Z by v_n = |s_n| for n ∈ Z. Clearly, v ∈ B and kvk_B =ksk_B. Moreover,

m

∑

λ^mk_v^−mk_B ≤ N

∑

m≥0

λ^mk_vk_B = ^N 1−λ

k_sk_B <+_∞.

Since B is complete, the series∑m≥0λ^mv^−m converges to some sequencex= (xn)_n∈Z ∈ B. It follows from the second property that

xn=

∑

m≥0

λ^m|sn−m|

for n ∈ _{Z. Since} |s¹_n| ≤ |xn| for n ∈ Z, we conclude that s¹ ∈ B and ks¹k_B ≤ kxk_B, which yields that the first inequality in (2.1) holds. One can show in a similar manner that s² ∈ B and that the second inequality in (2.1) holds.

Now let(X,k·k)be a Banach space and let k·k_n, for n ∈Z, be a sequence of norms onX such thatk·k_nis equivalent to k·k_{for each} n∈Z. For an admissible space B, let

Y_B= x= (x_n)_n∈Z⊂X: (kx_nk_n)_n∈Z∈ B . For x∈Y_B, we define

kxk_YB =k(kxnk_n)_n∈Zk_B. Proposition 2.5. (Y_B,k·k_YB)is a Banach space.

Proof. Let(x^k)_k∈Nbe a Cauchy sequence inY_B. Repeating arguments in the proof of Proposi- tion2.4, one can show that (x^k_n)_k∈N is a Cauchy sequence inXfor eachn∈_{Z. Let}

x_n= _lim

k→_∞x^k_n forn∈_Z and lets^k = (kx^k_nk_n)_n∈Z∈ Bfork∈_{N. Since}

kx^k_nk_n− kx^l_nk_n ≤ kx^k_n−x^l_nk_n forn∈_Z, we conclude that

ks^k−s^lk_B ≤ kx^k−x^lk_YB fork,l∈_N.

Hence,(s^k)_k∈N is a Cauchy sequence inB. Since Bis complete, it follows from property 2 in Proposition 2.4 that s^k → s in B whenk → _{∞, where} sn = kxnk_n for n ∈ Z. In particular, x = (x_n)_n∈Z ∈ Y_B. One can easily verify that the sequence (x^k−x)_k∈N converges to 0 in Y_B, which implies that(x^k)_k∈Nconverges toxinY_B.

3 Admissibility and exponential dichotomies

In this section we consider the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it in terms of the invertibility of a certain linear operator.

3.1 Basic notions

Let X be a Banach space and let L(X) be the set of all bounded linear operators from X to itself. Given a sequence(Am)_m∈Z inB(X), let

A(n,m) =

(A_n−1· · ·A_m ifn> m,

Id ifn= m. (3.1)

Definition 3.1. We say that (Am)_m∈Z admits an exponential dichotomy with respect to the se- quence of normsk·k_mif:

1. there exist projectionsP_m: X→ Xfor eachm∈_Zsatisfying

A_mP_m =P_m+1A_m form∈_Z (3.2)

such that each mapA_m|kerP_m: kerP_m→kerP_m+1 is invertible;

2. there exist constants D > 0 and 0< λ < 1 < µsuch that for eachx ∈ Xand n,m∈ _Z we have

k_A(n,m)Pmxk_n ≤Dλ^n−mkxk_m forn≥ m (3.3) and

k_A(n,m)Qmxk_n≤ Dµ^n−mkxk_m forn≤ m, (3.4) whereQ_m =Id−P_m and

A(n,m) = (_A(m,n)|kerPn)⁻¹: kerPm →kerPn

forn< m.

More generally, one can consider the notion of an exponential dichotomy for sequences of linear operators between different spaces. Namely, let Xn = (Xn,k·k), forn ∈Z, be pairwise isomorphic Banach spaces. Given a sequence of bounded linear operators A_m: X_m → X_m+1, form∈Z, one can defineA(n,m): X_m →X_nby (3.1) and introduce a corresponding notion of an exponential dichotomy, with projectionsPm: Xm →Xm form∈Z. All the results obtained in this section hold verbatim in this general setting, but we prefer avoiding the cumbersome notation.

Now let Bbe a Banach sequence space. Our main aim is to characterize the notion of an exponential dichotomy with respect to a sequence of norms in terms of the invertibility of the operatorT_B: D(T_B)⊂Y_B →Y_B defined by

(T_Bx)_n= xn−A_n−1x_n−1, n∈_Z, on the domainD(T_B)formed by all vectors x∈Y_B such thatT_Bx∈Y_B. Proposition 3.2. The linear operator T_B: D(T_B)⊂Y_B →Y_B is closed.

Proof. Let (_x^k)_k∈N be a sequence in D(T_B)converging to x ∈ Y_B such that T_Bx^k converges to y∈Y_B. It follows from the definition ofY_B and property 2 in Proposition2.4that

x_n−A_n−1x_n−1 = lim

k→_∞(x^k_n−A_n−1x^k_n−1) = lim

k→_∞(T_Bx^k)_n= y_n

forn∈Z, using the continuity of the linear operatorA_n−1. Therefore,x∈ D(T_B)andT_Bx=y.

This shows that the operatorT_B is closed.

Forx∈ D(TB)we consider the graph norm kxk⁰_Y

B =kxk_YB +kTxk_YB. Clearly, the operator

T_B: (D(T_B),k·k⁰_Y

B)→(Y,k·k_YB)

is bounded and from now on we denote it simply by TB. It follows from Proposition 3.2that (D(T_B),k·k⁰_Y

B)is a Banach space.

3.2 Characterization of exponential dichotomies

In this section we characterize the notion of an exponential dichotomy with respect to a se- quence of norms in terms of the invertibility of the operatorT_B.

Theorem 3.3. If the sequence(A_m)_m∈Zadmits an exponential dichotomy with respect to the sequence of normsk·k_m, then the operator T_B is invertible.

Proof. In order to establish the injectivity of the operator T_B, assume that T_Bx = _{0 for some} x∈ Y_B. Then xn = A_n−1x_n−1 forn∈ _{Z. Let}x^s_n= Pnxn andx_n^u= Qnxn. We havexn = x^s_n+x^u_n and it follows from (3.2) that

x^s_n= A_n−1x^s_n−1 and x^u_n= A_n−1x^u_n−1

forn∈ Z. Moreover,x^s_k =_A(k,k−m)x^s_k−m form≥0 and hence, kx_k^sk_k =k_A(k,k−m)x^s_k−mk_k

=k_A(k,k−m)P_k−mx_k−mk_k

≤Dλ^mkx_k−mk_k−m

≤ ^DN

α_B λ^mkxk_YB,

whereα_B = kχ_{0}k_B. Letting m →_∞ in the last term yields thatx^s_k = 0 for k ∈ Z. Similarly, x^u_k =_A(_k,k+_m)_x^u_{k+m form}≥0 and hence,

kx^u_kk_k = k_A(k,k+m)x_k^u_+mk_k

= k_A(k,k+m)Q_k+mx_k+mk_k

≤ Dµ^−mkx_k+mk_k+m

≤ ^DN

α_B µ^−mkxk_YB.

Therefore,x^u_k =0 fork∈_Zand hencex=0. This shows that the operatorTB is injective.

Now we show thatT_B is onto. Takey= (y_n)_n∈Z∈Y_B. For eachn∈_{Z, let} x¹_n=

∑

m≥0

A(n,n−m)P_n−my_n−m

and

x²_n=−

∑

m≥1

A(n,n+m)Q_n+my_n+m. We have

kx¹_nk_n≤

∑

m≥0

Dλ^mky_n−mk_n−m and kx²_nk_n≤

∑

m≥1

Dµ^−mky_n+mk_n+m.

It follows from property 3 in Proposition2.4that (x¹_n)_n∈Zand(x²_n)_n∈Z belong toYB. Now let x_n= x¹_n+x²_nforn∈ _Zandx= (x_n)_n∈Z. Thenx∈Y_B and one can easily verify that T_Bx =y.

This completes the proof of the theorem.

Now we establish the converse of Theorem3.3.

Theorem 3.4. If the operator T_B is bijective, then the sequence (A_m)_m∈Z admits an exponential di- chotomy with respect to the sequence of normsk·k_m.

Proof. For each n ∈ _{Z, let} X(n)be the set of all x ∈ X with the property that there exists a sequencex = (x_m)_m∈Z ∈ Y_B such that x_n = x and x_m = A_m−1x_m−1 for m> n. Moreover, let Z(n) be the set of all x ∈ X for which there exists z = (z_m)_m∈Z ∈ Y_B such that z_n = x and zm = A_m−1z_m−1form≤n. One can easily verify that X(n)andZ(n)are subspaces of X.

Lemma 3.5. For each n∈ _{Z, we have}

X=X(n)⊕Z(n). (3.5)

Proof of the lemma. Givenv ∈ X, we define a sequencey= (y_m)_m∈Zbyy_n= vandy_m =0 for m6=n. Clearly,y∈Y_B. Hence, there existsx∈Y_B such thatT_Bx=y, that is,

x_n−A_n−1x_n−1 =v (3.6)

and

x_m+1= A_mx_m form6=n−1. (3.7)

Sincex∈Y_B, we obtain

x_n∈X(n) and A_n−1x_n−1 ∈Z(n). Moreover, by (3.6), we have v∈X(n) +Z(n)_.

Now take v ∈ X(n)∩Z(n) and choose x = (x_m)_m∈Z and z = (z_m)_m∈Z in Y_B such that xn =zn =v,

x_m = A_m−1x_m−1 form> n and

z_m = A_m−1z_m−1 form≤n.

We definey= (ym)_m∈Zbyym = xm form≥ nandym =zm form<n. It is easy to verify that y∈Y_B andT_By=0. Since T_B is invertible, we havey=0 and thusy_n=v=0.

Let Pn: X → X(_n)_{andQn: X} → Z(_n)be the projections associated to the decomposition in (3.5).

Lemma 3.6. Property(3.2)holds.

Proof of the lemma. It is sufficient to show that

AnX(n)⊂X(n+1) and AnZ(n)⊂Z(n+1) forn∈_{Z. Take}v ∈X(n)andx= (x_m)_m∈Z∈Y_B such thatx_n=vand

xm = A_m−1x_m−1 form>n.

Then x_n+1 = A_nv ∈ X(n+1). Now takev ∈ Z(n)and choose z= (z_m)_m∈Z such that z_n = v and z_m = A_m−1z_m−1 for m ≤ n. We define z⁰ = (z⁰_m)_{m∈Z by} z⁰_m = z_m for m 6= n+_{1 and} z_n+1= Anv. Sincez⁰ ∈Y_B and

z⁰_m = A_m−1z⁰_m−1 form≤n+1, we conclude that Anv∈ Z(n+1).

Lemma 3.7. The linear operator A_n|kerP_n: kerP_n→kerP_n+1is invertible for each n∈ _Z.

Proof of the lemma. We first establish the injectivity of the operator. Assume that A_nv = 0 for v∈kerPn= Z(n)and choosez= (zm)_m∈Z ∈YB such thatzn= vand

z_m = A_m−1z_m−1 form≤n.

Moreover, we define y = (ym)_m∈Z by ym = 0 for m > n and ym = zm for m ≤ n. Clearly, y∈Y_B andT_By=0. Since T_B is invertible, we conclude thaty=0 and thusy_n=v=0.

In order to show that the operator is onto, takev∈kerP_n+1=Z(n+1)andz= (zm)_m∈Z∈ Y_B with z_n+1 = v andz_m = A_m−1z_m−1 for m ≤ n+1. Clearly, z_n ∈ Z(n) and A_nz_n = z_n+1. This shows that A_n|_kerP_nis onto.

Now we establish exponential bounds. Taken∈_Zandv∈ X. Moreover, letyandxbe as in the proof of Lemma3.5. For eachz≥1, we define a linear operator

B(z): (D(TB),k·k⁰_YB)→(YB,k·k_YB) by

(B(z)ν)_m =

(zν_m−A_m−1ν_m−1 ifm≤n,

1

zνm−A_m−1ν_m−1 ifm>n.

We haveB(₁) =T_B and

k(B(z)−T_B)νk_YB ≤ (z−1)kνk⁰_Y

B

forν ∈ D(T_B)andz ≥1. In particular, this implies that B(z)is invertible whenever 1 ≤ z<

1+1/kT_B⁻¹k, and

kB(z)⁻¹k ≤ ¹

kT_B⁻¹k⁻¹−(z−1)^.

Taket =1/zfor a givenz ∈(1, 1+1/kT_B⁻¹k)and letz∈Y_B be the unique element such that B(1/t)z=y. Writing

D⁰ = ¹

kT_B⁻¹k⁻¹−(1/t−₁)^, we obtain

kzk_YB ≤ kzk⁰_YB = kB(1/t)⁻¹_yk⁰_YB

≤ D⁰kyk_YB = ND⁰α_Bkvk_n

(where α_B = kχ_{0}k_B). For each m ∈ _{Z, let} x^∗_m = t^|m−n|−1zm and x^∗ = (x^∗_m)_m∈N. Clearly, x^∗ ∈Y_B. One can easily verify that T_Bx^∗ =yand hencex^∗ =x. Thus,

kx_mk_m =kx^∗_mk_m =t^|m−n|−1kz_mk_m

≤ ^N

α_Bt^|m−n|−1kzk_YB ≤ ^N

2D⁰

t t^|m−n|kvk_n ^(3.8) for m ∈ Z. Moreover, it was shown in the proof of Lemma 3.5 that Pnv = xn and Qnv =

−A_n−1x_n−1. Hence, it follows from (3.7) and (3.8) that

k_A(m,n)P_nvk_m= k_A(m,n)x_nk_m =kx_mk_m

≤ ^N

2D⁰

t t^m−nkvk_n ^(3.9)

form≥n. Similarly, it follows from (3.7) and (3.8) that k_A(m,n)Q_nvk_m ≤ ^N

2D⁰

t t^n−mkvk_n (3.10)

form < n. By (3.9) and (3.10), there exists D > 0 such that (3.3) and (3.4) hold takingλ = t andµ=1/t. This completes the proof of the theorem.

4 Nonuniformly hyperbolic sets

Now we consider an elaboration of the situation considered in Section 3. Namely, we char- acterize the notion of a nonuniformly hyperbolic set in terms of the invertibility of certain linear operators. More precisely, to each trajectory fⁿ(x) of a nonuniformly hyperbolic set of a diffeomorphism f one can associate a linear operator defined in terms of the sequence of tangent spaces d_fn(x)f (see the discussion after Definition 3.1). Moreover, each trajectory admits an exponential dichotomy with respect to the same sequence of tangent spaces and so it is natural to use arguments that are an elaboration of those in the former section.

4.1 Basic notions

Let M be a compact Riemannian manifold and let f: M→ Mbe aC¹ diffeomorphism.

Definition 4.1. An f-invariant measurable setΛ ⊂ M is said to be nonuniformly hyperbolicif there exist constants 0<λ<1<µand ad f-invariant splitting

TxM= E^s(x)⊕E^u(x)

forx ∈_Λsuch that given ε>0, there exist measurable functionsC,K: Λ→_R⁺such that for eachx ∈_Λ:

1. forv∈ E^s(x)andn≥0,

kdxfⁿvk_fn(x)≤C(x)λⁿe^εnkvk_x; (4.1) 2. forv∈ E^u(x)andn≥0,

kdxf⁻ⁿvk_f−n(x)≤C(x)µ⁻ⁿe^εnkvk_x; (4.2) 3.

∠(E^s(x),E^u(x))≥K(x); (4.3) 4. forn∈_Z,

C(fⁿ(x))≤C(x)e^ε|n| and K(fⁿ(x))≥ K(x)e^−ε|n|. (4.4) We note that a nonuniform hyperbolic set gives rise naturally to a parameterized family of exponential dichotomies with respect to a sequence of norms. More precisely, to each trajectory one can associate an exponential dichotomy (see [2]).

Proposition 4.2. Let Λ ⊂ M be a nonuniformly hyperbolic set. Then for each ε > ₀ such that λe^ε < 1 < µe^−ε there exists a normk·k⁰ = k·k^ε on T_ΛM such that for each x ∈ _Λthe sequence of linear operators

A_n =d_fn(x)f: T_fn(x)M →T_fn+1(x)M admits an exponential dichotomy with respect to the normsk·k⁰_fn(x).

Alternatively, Proposition4.2can be obtained as a consequence of the proof of Theorem4.3 below (the proof introduces a particular norm that is also adapted to our characterization of nonuniformly hyperbolic sets).

4.2 Characterization of nonuniformly hyperbolic sets

Given an admissible Banach sequence spaceB and a norm k·k⁰ on the tangent bundle T_ΛM, for eachx ∈ _Λwe denote byY_x the set of all sequencesµ = (µ_n)_n∈Z with µ_n∈ T_xnM, where xn = fⁿ(x), such that(kµnk⁰_xn)_n∈Z ∈ B. One can easily verify thatYx is a Banach space with the norm

k_µk=k(kµ_nk⁰_x

n)_n∈Zk_B. Finally, we define a linear operatorR_x by

(Rxµ)_n= µn−dx_n−1fµ_n−1, n ∈_Z, on the domain formed by allµ= (µn)_n∈Z∈Yx such thatRxµ ∈Yx.

Theorem 4.3. Let Λ ⊂ M be a nonuniformly hyperbolic set and let B be an admissible Banach sequence space. Then there existsε₀>0such that for everyε ∈(0,ε₀)there is a normk·k⁰ =k·k^ε on T_ΛM and a measurable function G: Λ→_R⁺such that for each x∈_Λ:

1. 1

2kvk_x ≤ kvk^ε_x ≤G(x)kvk_x, v∈ TxM; (4.5) 2.

G(fⁿ(x))≤e^2ε|n|G(x), n∈_Z; (4.6) 3. R_x:Y_x →Y_xis a well defined, bounded and invertible linear operator;

4. there exists a constant D>0(independent ofεand x) such that

kR⁻_x¹k ≤D. (4.7)

Proof. Since M is compact and f is continuous, there exists A > 0 such that kdxfk ≤ Aand kd_xf⁻¹k ≤Aforx ∈ M. Without loss of generality, one may assume that 1/A≤ λandµ≤ A (since otherwise one can simply increase A). Takeε₀ >0 such thatλe^ε0 <1<µe^−ε0. For each ε∈(0,ε₀)we introduce an adapted normk·k^ε on T_ΛM. Forv ∈E^s(x), let

kvk^ε_x =sup

n≥0

λ⁻ⁿe^−εnkdxfⁿvk_fn(x)

+sup

n<0

e^εnAⁿkdxfⁿk_fn(x)

. It follows from (4.1) that

kvk_x ≤ kvk^ε_x ≤(C(x) +1)kvk_x forv∈ E^s(x). (4.8) Moreover,

kdxf vk^ε_f(x)=sup

n≥0

λ⁻ⁿe^−εnkdxfⁿ⁺¹vk_fn+1(x)

+sup

n<₀

e^εnAⁿkdxfⁿ⁺¹vk_fn+1(x)

=λe^εsup

n≥0

λ⁻⁽ⁿ⁺¹⁾e^−ε(n+1)kd_xfⁿ⁺¹vk_fn+1(x)

+¹

Ae^−εsup

n<₀

Aⁿ⁺¹e^ε(n+1)kdxfⁿ⁺¹vk_fn+1(x)

≤λe^εkvk^ε_x

(4.9)

forv∈ E^s(x). Similarly, forv ∈E^u(x), let kvk^ε_x=sup

n≤0

µ⁻ⁿe^εnkd_xfⁿvk_fn(x)

+sup

n>0

A⁻ⁿe^−εnkd_xfⁿvk_fn(x)

. It follows from (4.2) that

kvk_x ≤ kvk^ε_x≤(C(x) +1)kvk_x forv∈ E^u(x). (4.10) Moreover,

kd_xf⁻¹vk^ε_f−1(x)=sup

n≤0

µ⁻ⁿe^εnkd_xfⁿ⁻¹vk_fn−1(x)

+sup

n>0

e^−εnA⁻ⁿkd_xfⁿ⁻¹vk_fn−1(x)

= ¹ µe^εsup

n≤0

µ⁻⁽ⁿ⁻¹⁾e^ε(n−1)kd_xfⁿ⁻¹vk_fn−1(x)

+¹

Ae^−εsup

n>0

e^−ε(n−1)A⁻⁽ⁿ⁻¹⁾kd_xfⁿ⁻¹vk_fn−1(x)

≤ ¹ µe^εkvk^ε_x

(4.11)

forv∈ E^u(x). One can show in a similar manner that

kd_xf vk^ε_f(x) ≤ A(e^ε+₁)kvk^ε_{x for}v∈ E^u(x)_{. (4.12)} For an arbitraryv∈ T_xM, we define

kvk^ε_x =max

kv^sk^ε_x,kv^uk^ε_x ,

wherev =v^s+v^uwithv^s∈ E^s(x)andv^u∈E^u(x). It follows from (4.3), (4.8) and (4.10) that 1

2kvk_x ≤ kvk^ε_x≤ ^C(x) +1

K(x) kvk_x forv∈T_xM.

Hence, (4.5) holds taking G(x) = (C(x) +₁)/K(x). Moreover, it follows from (4.4) that (4.6) holds. Finally, it follows from (4.9) and (4.12) that

kdxf vk^ε_f(x)≤ A(e^ε+1)kvk^ε_x (4.13) forx ∈_Λandv ∈T_xM.

Now letP(x): T_xM →E^s(x)andQ(x): T_xM →E^u(x)be the projections associated to the decompositionTxM =E^s(x)⊕E^u(x).

Lemma 4.4. There exists a constant Z>0(independent ofεand x) such that

kP(x)vk^ε_x ≤Zkvk^ε_x and kQ(x)vk^ε_x ≤Zkvk^ε_x (4.14) for x ∈_Λand v∈ TxM.

Proof of the lemma. For eachx∈ _Λlet

γ^ε_x =_infkv^s+v^uk^ε_{x :}kv^sk^ε_x =kv^uk^ε_x=_1,v^s∈ E^s(x)_,v^u ∈E^u(x) _.

Take a vectorv∈ T_xM such thatPv6=0 andQv 6=0, where P=P(x)andQ=Q(x). Then γ^ε_x ≤

Pv

kPvk^ε_x + ^Qv kQvk^ε_x

ε

x

= ¹

kPvk^ε_x

Pv+ kPvk^ε_x kQvk^ε_x^Qv

ε

x

= ¹

kPvk^ε_x

v+ kPvk^ε_x− kQvk^ε_x kQvk^ε_x ^Qv

ε

x

≤ ²kvk^ε_x kPvk^ε_x and thus,

kPvk^ε_x ≤ ² γ^ε_xkvk^ε_x

forv∈ TxM. In order to estimateγ^ε_x, takev^s∈E^s(x),v^u∈ E^u(x)such thatkv^sk^ε_x= kv^uk^ε_x =1.

It follows from (4.9), (4.11) and (4.13) (recall thatε<ε₀) that kv^s+v^uk^ε_x ≥ ¹

A(e^ε0+1)kd_xf(v^s+v^u)k^ε_f(x)

≥ ¹

A(e^ε0+1) kd_xf v^uk^ε_f(x)− kd_xf v^sk^ε_f(x)

≥ ¹

A(e^ε0+1)(µe^−ε0−λe^ε0) and thus,

γ^ε_x ≥ ¹

A(e^ε0+1)(µe^−ε0 −λe^ε0). Therefore, (4.14) holds taking

Z= ^2A(e^ε0+1) µe^−ε0−λe^ε0. This completes the proof of the lemma.

Now take x ∈ Λ. It follows from (4.13) that R_x is a well defined bounded linear operator on Y_x. We first show that it is onto. Let µ = (µ_n)_n∈Z ∈ Y_x. By Lemma 4.4, we have µ^s = (µ^s_n)_n∈Z∈Yx andµ^u = (µ^u_n)_n∈Z∈Yx, where

µ^s_n =P(fⁿ(x))µ_n and µ^u_n=Q(fⁿ(x))µ_n. For eachn∈_{Z, let}

ξ^s_n=

∑

m≥0

d_xn−mf^mµ^s_n−m and

ξ^u_n=−

∑

m≥1

dxn+mf^−mµ^u_n+m.

It follows from (2.1), (4.9), (4.11) and (4.14) (since ε< ε₀) thatξ^s = (ξ^s_n)_n∈Z andξ^u = (ξ^u_n)_n∈Z belong toYx. Moreover,

kξ^sk ≤ ¹

1−λe^ε0Zkµk and kξ^uk ≤ ¹

µe^−ε0 −1Zkµk

forn∈Z. Therefore,ξ = (ξ_n)_n∈Z, whereξ_n=ξ^s_n+ξ^u_n, belongs toY_xand kξk ≤Z

1

1−λe^ε0 + ¹ µe^−ε0−1

kµk. (4.15)

Moreover, one can easily verify that Rxξ = _µ.

Now we show thatRxis injective. Assume that Rxξ =0 for someξ = (ξn)_n∈Z∈Yx. Then ξ_n = d_xn−1f for n ∈ _Zand hence, ξ^s_n = d_xn−1fξ_n^s₋₁ and ξ^u_n = d_xn−1fξ^u_n−1 for n ∈ Z. For each k∈Z, it follows from (4.9) that

kξ_k^sk^ε_x

k ≤ (λe^ε)^mkξ^s_k−mk^ε_x

k−m ≤ ^NZ

α_B (λe^ε0)^mkξk

form≥0. Lettingm→+_∞, since λe^ε0 <1 we obtainξ^s_k =0. Similarly,ξ_k^u=0 for k∈_Zand thusξ =0. This shows thatRxis invertible. In addition, it follows from (4.15) that there exists a constant D> 0 (independent on xandε) such that (4.7) holds. This completes the proof of the theorem.

Now we establish the converse of Theorem4.3.

Theorem 4.5. LetΛ⊂ M be an f -invariant measurable set and let B be an admissible Banach sequence space. Assume that there exist D>0andε₀>0such that for eachε∈(0,ε₀)there is a normk·k^ε on T_ΛM and a measurable function G: Λ→_R⁺ such that for each x∈_Λ:

1. (4.5)and(4.6)hold;

2. R_x: Y_x →Y_x is a well defined bounded invertible linear operator and(4.7)holds.

ThenΛis a nonuniformly hyperbolic set.

Proof. Take x ∈ _Λ and v ∈ T_xM. We define µ = (µ_n)_n∈Z byµ₀ = v andµ_n = 0 for n 6= 0.

Clearly,µ∈Y_x. Now takeξ = (ξ_n)_n∈Z∈Y_x such thatR_xξ =µ. It can be written in the form ξ_n =

(d_xn−1fξ_n−1, n6=_0, d_x−1f−1+v, n=0.

We will show that v=v^s+v^u, wherev^s= ξ₀andv^u=−d_x−1fξ−1 is the hyperbolic splitting.

For eachz ≥1 we define an operatorB(z)_onY_x by (B(z)ν)_m =

(zν_m−d_xm−1fν_m−1 ifm≤_0,

1

zν_m−d_xm−1fν_m−1ν_m−1 ifm>0.

Clearly,

k(B(z)−R_x)νk ≤(z−1)kνk

forν∈Yx andz≥1. Therefore, B(z)is invertible whenever 1≤z <1+1/D, and kB(z)⁻¹k ≤ ¹

D⁻¹−(z−1)^.

Now takeλ∈ (0, 1)(independently onε) such thatλ⁻¹<1+1/Dand takeξ^∗ ∈Y_xsuch that B(λ⁻¹)_ξ^∗ =_{µ. Writing}

D⁰ = ¹

D⁻¹−(λ⁻¹−₁)^,