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arXiv:math/0408307v1 [math.DS] 23 Aug 2004

Liao Standard Systems and Nonzero Lyapunov Exponents for Di?erential Flows

Wenxiang Sun? School of Mathematical Sciences Peking University Beijing 100871, China sunwx@math.pku.edu.cn Todd Young Department of Mathematics Ohio University Athens 45701, Ohio, USA young@math.ohiou.edu fax: 740-593-9805 February 1, 2008

This paper is dedicated to the memory of Professor Liao Shantao, 1920–1997.

Abstract Consider a C 1 vector ?eld together with an ergodic invariant probability that has ? nonzero Lyapunov exponents. Using orthonormal moving frames along certain transitive orbits we construct a linear system of ? di?erential equations which is a reduced form of Liao’s “standard system”. We show that the Lyapunov exponents of this linear system coincide with all the nonzero exponents of the given vector ?eld with respect to the given probability. Moreover, we prove that these Lyapunov exponents have a persistence property that implies that a “Liao perturbation” preserves both sign and value of nonzero Lyapunov exponents.

Key Words and Phrases: Lyapunov exponent, standard linear system, Liao perturbation 2000 MSC: 37C15, 37A10, 34A26 Running title: Liao Systems and Lyapunov Exponents.

? Supported

partly by NNSFC 10171004. The ?rst author also thanks Ohio University for its hospitality during

the winter and spring of 2002 when this paper was written.

1

1

Introduction

Lyapunov exponents measure the asymptotic exponential rate at which in?nitesimally nearby points approach or move away from each other as time increases to in?nity. For a uniformly hyperbolic system with positive (resp. negative) Lyapunov exponents, its nearby system has positive (resp. negative) Lyapunov exponents as well. Using orthonormal frames moving along certain transitive orbits Liao (see [7]) constructed a system of linear equations, known as a “standard system”. In the hyperbolic case, our Main Theorem together with a result of Liao’s [7, Theorem 2.4.1] shows that the Lyapunov exponents of the standard system coincide with those of the original ?ow. Professor Liao had conjectured this result. For the complement of uniform hyperbolicity in the space of all C 1 systems with C 1 topology, understanding dynamics through Lyapunov exponents and SRB measures is incomplete but very important (see Palis [11]). Young [17, 18] constructed open sets of nonuniform hyperbolicity cocycles for certain special systems. In [16] Viana constructed an open set of systems with multidimensional nonhyperbolic attractors which have SRB measures [1]. For a compact surface, Bochi [2] showed that there is a residual set of C 1 area preserving di?eomorphisms so that each di?eomorphism in the set is either Anosov or has a zero Lyapunov exponent almost everywhere. In the 1960’s, Liao (see [7]) constructed a system of linear equations, known as a “standard system”. This system is essentially the variational equations along a typical orbit with respect to a typical orthonormal frame evolving along the orbit. Liao had used the standard system to give independent proofs of the C 1 closing lemma [7, Appendix A] and of the topological stability for Anosov ?ows [7, Chapter 2]. While Liao’s approach is obviously philosophically related to Lyapunov exponents, the connection has never been rigorously shown. In the hyperbolic case, our Main Theorem together with a result of Liao’s [7, Theorem 2.4.1] shows that the Lyapunov exponents of the standard system coincide with those of the original ?ow. Professor Liao had conjectured this result. We work with C 1 vector ?elds and develop a reduced form of Liao’s standard systems. We consider a C 1 vector ?eld together with an ergodic invariant probability that has ? nonzero Lyapunov exponents. Using typical moving orthonormal ?-frames along typical transitive orbits of the ergodic measure, and by using a characterization of the Lyapunov spectrum [5, 12] we construct a “reduced standard system” of di?erential equations and show that its Lyapunov exponents coincide with the nonzero exponents of the original vector ?eld. In the ?nal section we show that the nonzero Lyapunov exponents of the reduced standard system have certain persistence properties.

2

Now let us describe the main theorem of the present paper. We denote by M n a compact smooth n-dimensional Riemannian manifold and by S a C 1 di?erential system, or in other words, a C 1 vector ?eld on M n . As usual S induces a one-parameter transformation group φt : M n → M n , t ∈ R on the state manifold and therefore a one-parameter transformation group Φt = dφt : T M n → T M n , t ∈ R on the tangent bundle. A probability ν on M n is φ-invariant if it is φt -invariant for any t ∈ R. A φ-invariant probability is called φ-ergodic if every φ-invariant set has zero or full probability. For a compact metric space X and a topological ?ow ?t on it we denote by E (X, ?) the set of all φ-invariant and ergodic probabilities. Let ν be a φ-invariant and ergodic probability, i.e., ν ∈ E (M n , φ). From the Multiplicative Ergodic Theorem (see [4, 10]), there exists a φt -invariant subset B , with ν -full probability, such that for any x ∈ B and u ∈ Tx M n the following limit, called Lyapunov exponent, exists: λ := lim

t→∞

1 1 log Φt (u) ( or λ := lim log Φt (u) ). t→?∞ t t

(1.1)

It is known that ν has at most n di?erent Lyapunov exponents, where n indicates the dimension of the state manifold M n . Main Theorem Suppose that a φ-invariant and ergodic probability, ν ∈ E (M n , φ), has ? simple

nonzero Lyapunov exponents λ1 < λ2 < . . . < λ? , (1.2)

together with n ? ? zero Lyapunov exponents. Then the reduced standard linear system (de?ned in Section 4), dy = yA?×? (t), dt is well de?ned and has the following properties: 1. The matrix A?×? (t) is uniformly bounded and continuous with respect to t. 2. There exist u1 , u2 , . . . , u? ∈ R? such that 1 log y (t, ui ) = λi , t→∞ t lim where y (t, v ) denotes a unique solution of the initial value problem dy = yA?×? (t), dt 3. Consider a perturbation of the linear system dy = yA?×? (t) + f (t, y ), dt 3 sup

t∈R,y ∈R?

y ∈ R? ,

t ∈ R,

(1.3)

y (0, v ) = v.

(1.4)

f (t, y ) ≤ L < ∞,

(1.5)

? ? ? where f (t, y ) is Lipschitz in y . Then there exist u? 1 , u2 , . . . , u? ∈ R such that

1 log y (t, u? i ) = λi , t→∞ t lim where y (t, v ) denotes a unique solution of the initial value problem dy = yA?×? (t) + f (t, y ), dt y (0, v ) = v. (1.6)

In Section 2 we review frame bundles and the corresponding one parameter transformation groups induced by a given vector ?eld. In Section 3 we construct a reduced version of Liao’s “qualitative functions” and then use them to present a characterization of the Lyapunov spectrum. In Section 4 we construct the reduced standard linear system of ? di?erential equations on a given probability and establish a relation between the nonzero Lyapunov exponents of this probability and that of the linear system. We complete the proof of the Main Theorem in Section 5. An example in Section 5 illustrates that the original standard linear system of n di?erential equations introduced by Liao [7, Chapter 2] fails to satisfy the conclusions of the Main Theorem, and so, it is necessary to develop the reduced standard linear system of ? di?erential equations for the Main Theorem. In Section 7 we present the notion of Liao perturbation and point out by the Main Theorem that a certain type of perturbation, known as “Liao perturbation”, preserves the nonzero Lyapunov exponents.

2

One parameter transformation groups

We start from a C 1 vector ?eld S on a compact smooth n-dimensional Riemannian manifold M n , and its induced one-parameter transformation groups φt : M n → M n , t ∈ R on the state manifold and Φt = dφt : T M n → T M n , on the tangent bundle. Fix some integer ?, 1 ≤ ? ≤ n. Construct a bundle U? = ?ber over x is U? (x) = {(u1 , . . . , u? ) ∈ Tx M n × · · · × Tx M n : u1 , u2 , . . . , u? , are linearly independent}. (2.1)

x ∈M n

U? (x) of ?-frames, where the

Let p? : U? → M n denote the bundle projection. Denote by projk : U? → T M n the map which sends α ∈ U? to the k -th vector in α. The vector ?eld S induces a one-parameter transformation group on U? , which we denote (with the same notation as the tangent map for the sake of simplicity) by Φt , t ∈ R, namely, Φt (u1 , u2 , . . . , u? ) = (dφt (u1 ), dφt (u2 ), . . . , dφt (u? )). 4

For α = (u1 , u2 , . . . , u? ) ∈ U? and a nondegenerate ? × ? matrix B = (bij ) we write

? ? ?

α?B =

i=1

bi1 ui ,

i=1

bi2 ui , . . . ,

i=1

ai? ui

.

Then Φt (α ? B ) = Φt (α) ? B . By the Gram-Schmidt orthogonalization process there exists a unique upper triangular matrix Γ(α) with diagonal elements 1 such that α ? Γ(α) is orthogonal. Construct the bundle F? = Ux∈M n F? (x) of ?-orthogonal frames, where the ?ber over x is F? (x) = {(u1 , u2 , . . . , u? ) ∈ U? (x) | ui , uj = 0, 1 ≤ i = j ≤ ?}. (2.2)

The bundle projection is given by q? = p? |F? . The vector ?eld S then induces a one-parameter transformation group χt : F? → F? : α → Φt (α) ? Γ(Φt (α)). If we de?ne π : U? → F? by α → α ? Γ(α) then χt (α) = π (Φt (α)).

# # (x) of orthonormal ?-frames, where the ?ber over x is Construct a bundle F? = Ux ∈M n F? # F? (x) = {(u1 , u2 , . . . , u? ) ∈ F? (x) | ui = 1, i = 1, 2, . . . , ?}. # # Then F? is a compact metrizable space. Let π # : F? → F? be given by

(2.3)

(2.4)

π # (u1 , u2 , . . . , u? ) =

u1 u2 u? , ,..., u1 u2 u?

.

# # # # # Setting χ# t = π ? (χt |F? ), we get a one-parameter transformation group χt : F? → F? . Let # # # q? = q? |F? , then q? is a bundle projection. It is easy to check that the following properties hold:

q? ? χt = φt ? q? ,

# # q? ? χ# t = φt ? q? ,

# # χ# t ? π = π ? χt .

(2.5)

# Remark 2.1 We point out that the unitary ?-bundle U? of U? is not necessarily a compact metric m ∞ space. For instance, when ? = 2, there are sequences of 2-frames {αm = (um 1 , u2 )}m=1 such that the m angle between um 1 and u2 goes to zero as m → +∞. Such a sequence of frames has no accumulating # point inside U2 .

3

Qualitative functions

For α ∈ Fn , let ζαk (t) = projk χt (α) , k = 1, 2, . . . , n. Note that ζαk (t) > 0 for any t ∈ R. De?nition 3.1 For each k = 1, 2, . . . , n, we call ωk de?ned by: ωk : Fn → R : α → 5 dζαk (t) dt

t=0

(3.1)

# a qualitative function over the orthogonal n-frame bundle Fn and call ωk |Fn a qualitative function # over the orthonormal n-frame bundle Fn .

The qualitative function for vector ?elds was introduced by Liao in 1963, and it plays an important role in Liao theory [4-8]. Sun introduced its di?eomorphism version and described its relation with Lyapunov exponents in [13, 14], and determined in [15] the entropy of certain classes of Grassmann bundle systems by using these functions. From the de?nition it is easy to show that ωk (α) is continuous, ωk (χt (α)) = ωk (χ# t (α)) =

dζαk (t) 1 ζαk (t) dt , dζαk (t) , dt

and

so the following lemma is clear.

T 0

# Lemma 3.2 For α ∈ Fn and k = 1, 2, . . . , n, we have that log ζαk (T ) =

ωk (χ# t (α)) dt.

If we denote by Qν (M n , φ) the set of all points x ∈ M n that satisfy, for any continuous function f on M n , 1 t→±∞ t lim

0 t

f (φτ (x))dτ =

Mn

f d?,

(3.2)

# then Qν (M n , φ) is φ-invariant subset with ν -full probability. Similarly one can de?ne Q? (Fn , χ# ) # # for any probability ? ∈ E (Fn , χ# ) with q? ? (?) = ν .

The following is a slight modi?cation of [6, Theorem 4.1]. We state it here without proof.

# Lemma 3.3 For any given ? ∈ E (Fn , χ# ) and any permutation

r : {1, 2, . . . , n} → {r(1), r(2), . . . , r(n)}

# # # there exists ? ? ∈ E (Fn , χ# ) such that qn ?) and ? (?) = qn? (?

ωr(i) d? =

ωi d? ?, i = 1, 2, . . . , n.

Now we ?x the positive integer ?, ? ≤ n, as in the Main Theorem. De?ne

# # id? : Fn → F? : α = (v1 , . . . , vn?? , vn??+1 , . . . , vn ) → α ? = (vn??+1 , . . . , vn ). # Then, id? is a continuous projection. For α ? ∈ F? , set:

(3.3)

?k (? ζ α) = ζn??+k ? (id? )?1 (? α),

and,

ω ? k (? α) = ωn??+k ? (id? )?1 (? α).

(3.4)

?k (? It is clear by the de?nitions that both ζ α) and ω ? k (? α) are independent of the choice of preimages

# ?1 # ?k , ω , χ# ) set ? ? := id?? (?). Then in id? (α). Thus ζ ? k : F? → R are all well de?ned. For ? ∈ E (Fn # # ? ? ∈ E (F? , χ# ). Take α = (u1 , . . . , un ) ∈ Q? (Fn , χ# ). Then by Lemma 3.2,

1 1 log ζα(n??+k) (t) = lim t→±∞ t t→±∞ t lim

t 0

ωn??+k (χ# τ (α))dτ =

# Fn

ωn??+k d?,

6

for k = 1, 2, . . . , ?. Now write α ? := id? (α) = (un??+1 , . . . , un ). Then ω ? k (? α) = ω ? k (χ# α)) t (? α ?∈ =

# Q? ? (F? , χ ?αk dζ 1 ? (t) ?αk dt ζ ? (t) #

?αk dζ ? (t) dt t=0

and

?αk and ω and thus Lemma 3.2 holds for ζ ? k , k = 1, . . . , ?. Observe that for

) we have lim

t→∞

1 ?αk log ζ ? (t) = t =

# F?

ω ? k d? ? ωn??+k d? 1 log ζα(n??+k) (t), t k = 1, . . . , ?.

# Fn

= lim

t→∞

?k , F # → R is not necessarily the same as ζk : F # → R, We remark that the above function ζ ? ?

# # # and the function ω ? k : F? → R is not exactly the same as ωk : F? → R, where ζk , ωk : F? →R

are given in De?nition 3.1 with n replaced by ?. Proposition 3.4 Let ν ∈ E (M n , φ) be as in the Main Theorem, that is, it supports ? nonzero Lyapunov exponents λ1 < . . . < λ? together with n ? ? zero Lyapunov exponents. Then there exist

# # # two probabilities ? ∈ E (Fn , χ# ) and ? ? ∈ E (F? , χ# ), and two subsets Λ ? M n and W ? Fn such

that

# # 1. qn ?) = ν, id?? (?) = ? ?; ? (?) = ν, q?? (?

2. φt (Λ) = Λ,

χ# t (W ) = W,

ν (Λ) = 1,

and

?(W ) = 1;

# 3. For each x ∈ Λ and α ∈ W with qn (α) = x

t→±∞

lim

1 log ζα(n??+k) (t) = t

# Fn

ωn??+k d? =

# F?

ω ? k d? ? = λk ,

for k = 1, 2, . . . , ?. Proof. Take a φt -invariant subset Λ1 ? M n with ν -total probability so that at each point the spectrum of all Lyapunov exponents is λ1 , . . . , λ? together with n ? ? zeros. Furthermore, {λ1 , . . . , λ? 0, . . . , 0} =

# # ωk d? : ? ∈ E (Fn , χ# ), qn ? (?) = ν, k = 1, 2, . . . , n .

The existence of Λ1 follows from the hypothesis of the present proposition and Theorem 2.2 in [12].

# # Choose an arbitrary ?1 ∈ E (Fn , χ# ) to cover ν , i.e., qn ? (?1 ) = ν . We claim that

{λ1 , . . . , λ? , 0, . . . , 0} =

ωk d?1 , k = 1, 2, . . . , n .

(3.5)

# # # , χ# )) = 1, thus , χ# )) = 1 and ν (qn Q?1 (Fn Observe that ?1 (Q?1 (Fn # # , χ# ) ν qn Q?1 (Fn

Λ1 = 1.

7

# # , χ# ) Take x ∈ qn Q?1 (Fn

# Λ1 and α = (u1 , . . . , un ) ∈ Fn (x)

# , χ# ). Remember that ωk Q?1 (Fn

is a continuous function. By Lemma 3.2 we then have 1 1 log ζαk (t) = lim t→±∞ t t→±∞ t lim

t 0

ωk (χ# s (α))ds =

ωk d?1 ,

k = 1, 2, . . . , n.

We point out that in the case when index k = 1 we have 1 1 log Φt (u1 ) = lim log ζα1 (t) t→±∞ t t→±∞ t lim = ω1 d?1 .

If we suppose that Equation (3.5) is not true, then there would exist a minimal index i0 > 1 such that

t→±∞

lim

1 log Φt (ui0 ) = t

ωk d?1 ,

for all k = 1, 2, . . . , n.

# n Note that {proj1 χ# t (α), . . . , projn χt (α)} is an orthonormal frame on the tangent space Tφt (x) M

and < Φt (ui0 ), projj χ# t (α) >= 0 for each j = i0 + 1, . . . , n. We can represent

Φt (ui0 ) Φt (ui0 )

as

Φt (ui0 ) # # = a1 (t) proj1 χ# t (α) + a2 (t) proj2 χt (α) + . . . + ai0 (t) proji0 χt (α), Φt (ui0 ) where |ak (t)| ≤ 1, k = 1, 2, . . . , i0 . Now let us suppose that limt→∞ |ai0 (t)| > 0. Observe that both ai0 (t) Φt (ui0 ) proji0 χ# t (α) and proji0 χt (α) express the same projection of Φt (ui0 ) on the direction determined by proji0 χ# t (α), thus |ai0 (t)| Φt (ui0 ) = ζαi0 (t). Therefore by Lemma 3.2

t→∞

lim

1 1 log Φt (ui0 ) = lim sup log Φt (ui0 ) t t→∞ t 1 1 = lim sup log |ai0 (t)|?1 + lim sup log ζαi0 (t) t→∞ t t→∞ t 1 = lim log ζαi0 (t) t→∞ t 1 t ωi0 (χ# = lim s (α)) ds t→∞ t 0 = ωi0 d ?1 .

This is a contradiction to the choice of i0 . For the case limt→∞ |ai0 (t)| = 0, one then gets that lim 1 log Φt (ui0 ) t 8

t→∞

coincides with (3.5) holds.

ωi d?1 for some i < i0 , again a contradiction to the choice of i0 . Consequently,

Now there is a permutation r : {1, 2, . . . , n} → {r(1), r(2), . . . , r(n)} so that ωr(i) d?1 = 0, i = 1, 2, . . . , n ? ?, and ωr(i) d?1 = λi?(n??) , i = n ? ? + 1, n ? ? + 2, . . . , n.

# # # From Lemma 3.3, there exists a covering probability ? ∈ E (Fn , χ# ) of ν , qn ? (?) = qn? (?1 ) = ν , so

that

ωi d? =

ωr(i) d?1 , i = 1, 2, . . . , n.

# qn (W ).

# De?ne W := Q? (Fn , χ# ) and Λ := Λ1

Then ν (Λ) = ?(W ) = 1, χ# t (W ) =

# # W, φt (Λ) = Λ, t ∈ R, and qn (W ) = Λ. De?ne ? ? := id?? (?). Then ? ? ∈ E (F? , χ# ). Clearly # q? ?) = ν , and ? (?

ω ? k d? ?= Take x ∈ Λ and α ∈ W lim

# Fn (x), then

ωn??+k d?, k = 1, . . . , ?.

t→±∞

1 log ζα(n??+k) (t) = t

ωn??+k d? =

ω ? k d? ?, k = 1, 2, . . . , ?.

This completes the proof of Proposition 3.4.

# # # Corollary 3.5 For any given α ? = (un??+1 , . . . , un ) ∈ Q? α) ∈ Λ we have ? (F? , χ ) with q?? (?

t→?∞

lim

1 log Φt (ui ) < 0, t 1 log Φt (ui ) > 0, t

i = n ? ? + 1, . . . , n ? ? + p,

and,

t→+∞

lim

i = n ? ? + p + 1, . . . , n,

where p satis?es: λ1 < . . . < λp < 0 < λp+1 < . . . < λ? . Proof. For n ? ? + p + 1 ≤ i0 ≤ n, we have by Lemma 3.2 that, 0<

# Fn

ωi0 d?

1 t ωi0 (χ# τ (α))dt t→+∞ t 0 1 log ζαi0 (t) = lim t→+∞ t 1 ≤ lim log Φt (ik0 ) . t→∞ t = lim For n ? ? + p + 1 ≤ i0 ≤ n, we may deduce a similar inequality. 9

4

Reduced standard linear systems of ? di?erential equations

We start this section from the φ-invariant, ergodic probability ν ∈ E (M n , φ) assumed in the Main

# # Theorem together with its two covering probabilities ? ∈ E (Fn , χ# ) and ? ? ∈ E (F? , χ# ) and the # corresponding total probability subsets Λ ? M n and W ? Fn as in Proposition 3.4. Take a point

x ∈ Λ and an orthonormal frame α ∈ W

# Fn (x). Then

1 1 log ζαk (t) = lim t→∞ t t→∞ t lim = =

# Fn

t 0

ωk (χ# τ (α))dτ

ωk d? ω ? k?(n??) d? ?, k = n ? ? + 1, . . . , n.

# F?

In this section we will construct the reduced standard linear system needed in the Main Theorem

# (x), by developing along the orbit orb(x, φ) with respect to the given orthonormal frame α ∈ Fn

the technique in [4].

n Since χ# t (α) is an orthonormal frame at Tφt (x) M , there exists an n × n matrix Bα (t) such that

Φt (α) = χ# t (α) ? Bα (t). De?ne Rα(t) =

dBα (t) dt

? Bα (t)?1 . De?ne a diagonal matrix

ζα (t) = diag ζα1 (t), ζα2 (t), . . . , ζαn (t) .

# ?1 From Gram-Schmidt orthogonalization, χ# t (α) = Φt (α) ? Γ(Φt (α)) ? ζα (t), or, Φt (α) = χt ? ζα (t) ?

Γ(Φt (α))?1 , where Γ(Φt (α)) is an n × n upper triangular matrix with elements 1 on the diagonal. So Bα (t) = ζα (t) ? Γ(Φt (α))?1 , which is di?erentiable with respect to t ∈ R. Observe 1 dζαk (t) = ωk (χ# t (α)), k = 1, . . . , n, ζαk (t) dt and dBα (t) ? Bα (t)?1 dt ?

dζα1 (t) 1 ζα1 (t) dt dζα2 (t) 1 ζα2 (t) dt

? ..

Thus

? ? ? =? ? ? ? ω1 (χ# t (α))

? .

dζαn (t) 1 ζαn (t) dt

? ? ? ?. ? ? ?

? ? ? Rα (t) = ? ? ? ?

?

? ω2 (χ# t (α)) .. . ωn (χ# t (α)) 10

?

? ? ? ?. ? ? ?

(4.1)

Now let us denote Rα (t)T by (rij (t))n×n , where rij (t) = 0 if i < j ; rii (t) = ωi (χ# t (α)), i, j =

1 1, . . . , n. Set α ? := id? (α). Recall from Section 3 that, ω ? i (? α) = ω(n??)+i ? id? α), for i = 1, . . . , ?. ? (?

We de?ne a triangular ? × ? matrix A?×? (t) = (aij (t))?×? as follows: aij (t) = 0 if i < j ; aij (t) = r(n??+i)(n??+j ) (t) if i > j ; aii (t) = ω ? i (χ# α)), i, j = 1, . . . , ?. Thus t (? ? ω ? 1 (χ# α)) t (? ? ? ω ? 2 (χ# α)) ? t (? ? A?×? (t) = ? .. ? . ? α)) ? ω ? ? (χ# t (? where α ∈ W

# Fn (x) and x ∈ Λ.

?

? ? ? ?, ? ? ?

(4.2)

De?nition 4.1 We call dy = yA?×? (t), dt respect to an orthonormal n-frame α, where A?×? (t) is given by (4.2). Proof of the Main Theorem (1.)(2.). For (1) it is su?cient to show A?×? (t) is uniformly bounded. In [4] Liao proved that supt∈R Rα (t) < ∞, from which it is easy to get sup A?×? (t) ≤ sup Rα (t) < ∞.

t∈R t∈R

(4.3)

the reduced standard linear system of ? di?erential equations for the given system (M n , S, ν ) with

Now we prove the Main Theorem (2.) by showing the following proposition. Proposition 4.2 Let ν ∈ E (M n , φ) be as in the Main Theorem. Let us take covering probabilities

# # # # ? ∈ E (Fn , χ# ), and ? ? ∈ E (F? , χ# ), satisfying qn ?) = ν , and take a ?-total probability ? (?) = q?? (? # subset W ? Fn and a ν -total probability subset Λ ? M n as in Proposition 3.4. Take x ∈ Λ and

α∈W

# Fn (x) and construct the reduced standard linear system (4.3) of ? di?erential equations

as in De?nition 4.1. For a coordinate vector ei = (0, . . . , 0, 1(i), 0, . . . , 0) ∈ R? denote by y ?(t, ei ) a unique solution of the initial value problem (1.4) with y (0, ei ) = ei . Then

t→±∞

lim

1 log y ?(t, ei ) = λi , t

i = 1, . . . , ?.

Proof. Solving the initial value problem dy? =ω ? ? (χ# α))y? , t (? dt we get y? (t, e? ) = e? e

t 0

y (0) = e? ,

ω ? ? (χ# ? ))dτ τ (α

.

11

This equality together with Proposition 3.4 implies the following 1 1 log |y? (t, e? )| = lim t→∞ t t→∞ t 1 = lim t→∞ t lim = =

# Fn

t 0 t 0

ω ? ? (χ# α))dτ τ (? ωn (χ# τ (α))dτ

ωn d? ω ? ? d? ?

# F?

= λ? . Solving the initial value problem dy??1 =ω ? ??1 (χ# α))y??1 + e? r?(??1) (t)e t (? dt we get y??1 (t, e??1 ) = e??1 e + e? e Thus, 1 1 log |y??1 (t, e? )| = lim t→∞ t t→∞ t 1 = lim t→∞ t lim = =

# Fn t 0 t 0

ω ? ? (χ# ? ))dτ τ (α

,

y??1 (0, e??1 ) = e??1

ω ? ??1 (χ# ? ))dτ τ (α t

τ 0

t 0

ω ? ??1 (χ# ? ))dτ τ (α 0

r?(??1) (τ )e

ω ? ? (χ# ? ))ds ? s (α

e

τ 0

ω ? ??1 (χ# ? ))ds s (α

dτ.

t 0 t 0

ω ? ??1 (χ# α))dτ τ (? ωn?1 (χ# τ (α))dτ

ωn?1 d? ω ? ??1 d? ?

# F?

= λ??1 . By repeating this procedure we will obtain: lim 1 log |yj (t, ej )| = λj , t

t→∞

for j = 1, . . . , ?. From the form of the functions y? (t, e? ), . . . , y1 (t, e1 ), which depend linearly on the initial values e1 , e2 , . . ., e? , we get easily y ?(t, ei ) = |yi (t, ei )|.

12

Therefore 1 1 log y ?(t, ei ) = lim log |yi (t, ei )| t→∞ t t→∞ t lim = λi . This proves the proposition and thus proves parts (1.) and (2.) of the Main Theorem.

5

Proof of the Main Theorem (3.)

# Let ν ∈ E (M n , φ) denote the given probability in the Main Theorem. Let ? ∈ E (Fn , χ# ) and

In this section we will complete the proof of the Main Theorem.

# # ? ? ∈ E (F? , χ# ) be the covering probabilities, and let Λ ? M n and W ? Fn be the two total # probability sets, qn (W ) = Λ, as in Proposition 3.4. Write ?i (? ?) =

# F?

ω ? i d? ?, i = 1, 2, . . . , ?. Then

we have that ? satis?es: ?1 (? ?) = λ1 < ?2 (? ?) = λ2 < . . . < ?? (? ?) = λ? . Let T1 ≥ 1 be a ?xed constant and let Ti+1 = 2Ti , i = 1, 2, . . . . Recall from Section 3 the projection map (3.3). De?nition 5.1 For η > 0 we denote by D(?, η ) the set of all γ ? ∈ id? (W ) with the property that for each integer i ≥ 1 there exist an integer c = c(? γ , i, η ) ≥ i and a sequence . . . < s(?2) < s(?1) < s(0) = 0 < s(1) < s(2) < . . .

j →?∞

lim s(j ) = ?∞,

j →+∞

lim s(j ) = +∞,

(5.1)

such that

1 1 max ?k (? ?) ? l τ =0 k=1,2,...,? δTc

l? 1

(τ +1)δTc τ δTc

ω ? k (χ# γ ))dt < η, t+s(j )Tc (?

l = 1, 2, . . . ; j = 0, ±1, ±2, . . . ; δ = ±1. Lemma 5.2 ? ?(D(?, η )) > 0. Proof. This is a partial result of [8, Theorem 2.1], where Liao gave a general proof. We present a proof of our case here for convenience to readers. Set hk (? γ , T, δ ) = |?k (? ?) ? h(? γ , T, δ ) =

k=1,2,...,?

1 δT

δT 0

ω ? k (χ# γ ))dt|, t (?

# 0 < T < +∞, δ = ±1, γ ? ∈ F? ,

max

hk (? γ , T, δ ). 13

When γ ? ∈ id? (W ) we choose and ?x γ ∈ W with id? (γ ) = γ ? . We have from Proposition 3.4, for k = 1 , 2 , . . . , ?, 1 T →±∞ T lim

T 0 T 0

ω ? k (χ# γ ))dt = lim t (?

1 T →±∞ T

ωn??+k (χ# t (γ ))dt

= ?k (? ?) = λk . By Chapter 6 in [9], lim ?k (? ?) ? 1 δT

a+T a

T →∞

# F?

ω ? k (χ# γ ))dt d? ? = 0, t (?

where the convergence is uniform with respect to the choice of a ∈ R. For δ = 1 and a = 0 we get

T →∞

lim

# F?

hk (? γ , T, +1) d? ? = 0, k = 1, . . . , ?.

For δ = ?1, taking a = ?T , we get

T →∞

lim

# F?

hk (? γ , T, ?1) d? ? = 0,

k = 1, . . . , ?.

Therefore

T →∞

lim

# F?

h(? γ , T, δ )d? ? = 0,

δ = ±1.

For η > 0 one can thus take an integer d = d(η ) > 0 such that

# F?

h(? γ , Td , δ )d? ?<

η , δ = ±1. 30

# # Let us consider a ? ? -preserving homeomorphism ρ = χ# Td : F? → F? . Applying the Birkho?

Ergodic Theorem to the homeomorphism ρδ and continuous function h(? γ ; T, δ ), δ = ±1, there is a

# ? ?-measurable subset X ? F? with ? ?(X ) = 1 such that for any γ ? ∈ X, δ = ±1, the following limit

exists lim

1 h(ρδτ (? γ ), Td , δ ) = h? (? γ , Td , δ ). l→∞ l τ =0 Moreover,

# F?

l? 1

h? (? γ , Td , δ )d? ?=

# F?

h(? γ , Td , δ )d? ?<

η 2}

η . 30

# This implies that the set {γ ? ∈ F? | h? (? γ , Td , δ ) >

is ? ? -measurable and has ? ?-probability less

than or equal to

1 12 ,

δ = ±1. Applying Egoro?’s Theorem (see, for example, [3]) there exists a

3 4

subset Y of X with ? ?(Y ) ≥

> 0 such that

l? 1

1 h(ρδτ (? γ ), Td , δ ) < η, l τ =0 14

?γ ? ∈ Y, l ≥ ? l.

Therefore

1 1 max ?k (? ?) ? l τ =0 k=1,2,...,? δTd or 1 1 max ?k (? ?) ? l τ =0 k=1,2,...,? δTd

l? 1

l? 1

δTd 0

ω ? k (χ# γ ))dt < η, t+δτ Td (?

δ (τ +1)Td δτ Td

ω ? k (χ# γ ))dt < η, ?γ ? ∈ Y, l ≥ ? l. t (?

From the Poincar? e Recurrence Theorem, let us take a subset Y ′ of Y , ? ?(Y ′ ) = ? ?(Y ), with the property that for each γ ? ∈ Y ′ there exists a sequence {s(j )} of the form (5.1), so that ρs(j ) (? γ) ∈ Y . This gives rise to 1 1 max ?k (? ?) ? k =1 , 2 ,...,? l τ =0 δTd l≥? l,

l? 1 δ (τ +1)Td

ω ? k (χt+s(j )Td (? γ ))dt < η,

δτ Td

j = 0, ±1, ±2, . . . ,

δ = ±1.

# Denote by ξ (γ ) a character function for Y on F? . Let us consider a ? ?-preserving homeomorphism # # ψ : F? → F? .

Set

1 ?(? ξ (ψ δτ (? γ )), ξ γ , ψ, δ ) := lim sup l l→∞ τ =0 ? is a Baire function. Let Then ξ

l? 1

# γ ? ∈ F? , δ = ±1.

# ? E (η, ψ, δ ) = {γ ? ∈ F? | ξ (? γ , ψ, δ ) > 0},

E (η, ψ ) = E (η, ψ, ?1)

E (η, ψ, +1).

# By the Birkho? Ergodic Theorem there exists a subset Z ? F? , ? ?(Z ) = 1, such that for all

γ ?∈Y

Z , the limit exists 1 ξ (ψ δτ (? γ )) = ξ ? (? γ , ψ, δ ). l→∞ l τ =0 lim

l? 1

Since ?(? 1≥ξ γ , ψ, δ ) ≥ ξ ? (? γ , ψ, δ ) ≥ 0, then ?(E (η, ψ, δ )) ≥ ≥ ?(? ξ γ , ψ, δ )d? ? ξ ? (? γ , ψ, δ )d? ? Z)

=? ?(Y =? ?(Y ) ≥ 3 , 4

15

1 for both δ = 1 and δ = ?1, which implies then ? ?(E (η, ψ )) ≥ 2 .

Now for each integer i ≥ 1 take ψ as ψi = χ# Tc where T = Tc , c = c(i, η ) ≥ 1 with 2c(i,η)?d ≥ ? l. Moreover we take c(i, η ) < c(i + 1, η ), i = 1, 2, 3, . . . . Then Tc(i,η) = 2c(i,η)?d Td ≥ ? lTd . Write F (η, ψi ) = Y ′ namely,

s(j ) γ) ψi (? l? 1 1 . E (η, ψi ) then ? ?(F (η, ψi )) ≥ 4 s(j )

Take γ ? ∈ F (η, ψi ). Then there is a sequence {s(j )} of the form (5.1) such that ξ (ψi ∈ Y, j = 0, ±1, ±2, . . .. Recall by de?nition Tc = 2c?d Td . We have

δ (τ +1)Tc δτ Tc

(? γ )) = 1,

1 1 max ?k (?) ? l τ =0 k=1,2,...,? δTc

l? 1

ω ? k (χ# γ ))dt t+s(j )Tc (?

δ (τ +1)Tc δτ Tc

=

1 1 max ?k (?) ? l τ =0 k=1,2,...,? δTc 1 l 2 c ?d 1 l 2 c ?d 1 l 2 c ?d

l2c?d ?1 τ =0 l2c?d ?1 τ =0 l2c?d ?1 k=1,2,...,? k=1,2,...,?

ω ? k (χ# t (ψi

δ (τ +1)Td

s(j )

(? γ )))dt

s(j )

≤

max

?k (?) ?

1 δTd 1 δTd

δτ Td δTd 0

ω ? k (χ# t (ψi

(? γ )))dt

=

max

?k (?) ?

δτ ω ? k (χ# t (ρ (ψi

s(j )

(? γ )))dt

=

h(ρδτ (ψi

τ =0

s(j )

(? γ )), Td , δ )

< η, where γ ? ∈ F (η, ψi ); l = 1, 2, . . ., j = 0, ±1, ±2, . . ., and δ = ±1. Letting D(?, η ) := F (η, ψi ) we complete the proof of Lemma 5.1. Corollary 5.3 Set F (η ) =

i=1,2,... F (η, ψi ),

where F (η, ψi ) is as in the proof of Lemma 5.2.

1 4

Then F (η ) is a Borel subset. Since ? ?(F (η, ψi )) ≥ F (η, ψi+1 ), i = 1, 2, . . ., we then have ?(F (η )) ≥ Theorem 5.4 (Liao, [8]) Consider two systems dy = yC (t) + f (t, y ), dt

and c(i, η ) < c(i + 1, η ), and thus F (η, ψi ) ?

1 > 0. 4

(t, y ) ∈ R × R?

(5.2)

dy = yC (t) (t, y ) ∈ R × R? . dt Let the following (i)(ii) and (iii) hold. 16

(5.3)

(i). For any t ∈ R, C (t) = (cij )?×? is a lower triangular ? × ? matrix. C (t) is continuous with respect to t and uniformly bounded. (ii). There exist constants λ > 0, T > 0, c =

1 16

min{1, λ}, and a bi-in?nite sequence {s(j )} of

the form (5.1) so that for some integer p ∈< 0, ? > the following inequalities hold ?λ < 1 1 max{?λ, max k=1,2,...,p δT l τ =0

l? 1 l? 1 δ (τ +1)T

ckk (t + s(j )T )dt} < ?λ + c,

δτ T δ (τ +1)T

λ?c<

1 1 min{λ, min k=p+1,...,? δT l τ =0

ckk (t + s(j )T )dt} < λ

δτ T

j = 0, ±1, ±2, . . . ; l = 1, 2, . . . ; δ = ±1. (iii). Vector function f(t, y) is continuous with (t, y ) and is uniformly bounded and Lipschitz with respect to y . Then, for each u? ∈ R? there exists uniquely u ∈ R? so that the solutions y (t, u? ) and y (t, u) of the initial value problem (5.2), (5.3) with initial conditions y (0; u? ) = u? and y (0; u) = u, respectively, satisfy the following relation. (a). There is a integer sequence . . . < m(?2) < m(?1) < m(0) = 0 < m(1) < m(2) < . . . lim m(j ) = ?∞, lim m(j ) = +∞

j →?∞

j →+∞

so that sup y (m(k )T, u) ? y (m(k )T, u? ) < ∞.

k ∈Z

(b). The map ?? : R? → R? , u? → u is surjective. (c). There exist constants C ? > 0 and d > 0 so that y (t, ?? (u? )) ? y (t, u? ) ≤ C ? exp(2c|t ? s(j )T | + d), Proof. (a) and (b) are Theorem 3.1 in [8], its Corollary 1 is (c). Proof of the Main Theorem (3.). For ν ∈ E (M n , φ) let us consider all its ? nonzero Lyapunov exponents λ1 < . . . < λp < λp+1 < . . . < λ? , where λp < 0 < λp+1 . We recall again from

# # # # Proposition 3.4 the covering probabilities ? ∈ E (Fn , χ# ), ? ? ∈ E (F? , χ# ), qn ?), ? (?) = ν = q?? (? # # (W ) = Λ. And consider continuous functions and the subsets W ? Fn and Λ ? M n with qn # ?αk , ω ζ ? k : F? → R as in Section 3. Take an arbitrary positive real λ with λp < λ < λp+1 and

j = 0, ±1, ±2, . . . .

λ<

1 min {|λi ? λj |, |λi ? 0|}. 2 1≤i=j ≤? 17

Write c :=

1 16

min{1, λ} as in Theorem 5.4 and write η :=

c 2

as in Lemma 5.2. We take and ?x an

orthonormal ?-frame α ? ∈ F (η ), where F (η ) is de?ned in the Corollary 5.3. Recall by construction F (η ) ? id? (W ), one can take α ∈ W with id? (α) = α ? . By using the moving orthonormal n-frame { χ# t (α); t ∈ R} we can construct as in Section 4 a reduced standard linear system (4.3) of ? di?erential equations. As in Section 4 we can prove the Main Theorem(i)(ii) with respect to this linear system of di?erential equations. Now let us consider a perturbed system (1.5) where f (t, y ) is Lipschitz and uniformly bounded. Observe that the kk -th entry of the matrix A?×? (t) is akk (t) = ω ? k (χ# α)), t (? k = 1, 2, . . . , ?.

Since α ? ∈ F (η ) and p ≤ ? there exist, by Lemma 5.2 and Corollary 5.3 a positive number T > 0 and a sequence {s(j )} of the form (5.1) such that 1 1 ?) ? max ?k (? k =1 , 2 ,...,p l τ =0 δT

l? 1 (τ +1)δT τ δT

ω ? k (χ# γ ))dt < η, t+s(j )T (?

l = 1, 2, . . . ; j = 0, ±1, ±2, . . . ; δ = ±1. Observe λk = ?k (? ?) < λ, and so we get

1 l τ =0

l? 1

1 k=1,2,...,p δT max

(τ +1)δT τ δT l? 1

ω ? k (χ# γ ))dt ? λ t+s(j )T (?

(τ +1)δT τ δT (τ +1)δT τ δT

=

1 1 max l τ =0 k=1,2,...,p δT 1 1 max l τ =0 k=1,2,...,p δT

l? 1 l? 1

ω ? k (χ# γ ))dt ? λ t+s(j )T (? ω ? k (χ# γ ))dt ? λk t+s(j )T (?

(τ +1)δT τ δT

≤

≤

1 1 max ?k (? ?) ? l τ =0 k=1,2,...,p δT

ω ? k (χ# γ ))dt t+s(j )T (?

≤η c = . 2 18

Therefore ?λ < 1 1 max{?λ, max k=1,2,...,p δT l τ =0

l? 1 δ (τ +1)T δτ T

ω ? k (χ# t+s(j )T (γ ))dt} < ?λ + c,

for j = 0, ±1, ±2, . . . ; l = 1, 2, . . . ; δ = ±1. Similarly, λ?c< 1 1 min{λ, min k=p+1,...,? δT l τ =0

l? 1 δ (τ +1)T δτ T

ω ? k (χ# t+s(j )T (γ ))dt} < λ,

for j = 0, ±1, ±2, . . . ; l = 1, 2, . . . ; δ = ±1. Now we apply Theorem 5.4 to complete the Main Theorem. Since ?? in Theorem 5.4 is surjective, for uk = (0, . . . , 0, 1(k ), 0, . . . , 0) ∈ R?

? ? ? there exist u? k ∈ R so that ? (uk ) = uk , k = 1, . . . , ?. From Theorem 5.4 the solution y (t, uk ) of

the initial value problem dy = yA?×? (t), dt and the solution y (t, u? k ) of the initial value problem dy = yA?×? (t) + f (t, y ), dt satisfy the relation

? y (t, u? k ) ? y (t, uk ) ≤ C exp(2c(|t ? s(j )T | + d)) ? y (0, u? k ) = uk

y (0, uk ) = uk

for some constants C ? > 0 and d > 0. Letting j = 0 and thus s(j ) = 0 it follows

? y (t, u? k ) ≤ y (t, uk ) + C exp(2c|t| + d)

≤ y (t, uk ) × C ? exp(2c|t| + d) ? > 0. This yields by Proposition 4.2 for |t| ≥ t lim sup

t→∞

1 1 log y (t, u? log y (t, uk ) + 2c k ) ≤ lim sup t t→∞ t = λk + 2c,

where we recall c =

1 16

min{1, λ}. Since λ and thus c can be taken small enough, we get lim sup 1 log y (t, u? k ) ≤ λk , t k = 1, 2, . . . , ?.

Now one can easily get

? y (t, uk ) ≤ y (t, u? k ) × C exp(2c|t| + d)

19

? > 0. This gives rise to for |t| ≥ t λk = lim inf

t→∞

1 log y (t, uk ) t 1 ≤ lim inf log y (t, u? k ) + 2c. t→∞ t

Thus λk ? 2c < lim inf 1 log y (t, u? k) t→∞ t 1 ≤ lim sup log y (t, uk ) t→∞ t

< λk + 2c. Since c can be taken small enough, we get lim 1 log y (t, u? k ) = λk , k = 1, 2, . . . , ?. t

t→∞

This completes the proof of the Main Theorem. Example. When ? < n ? 1, the system (M n , φ, ν ) is not hyperbolic. In this case the Main Theorem does not hold for the linear system [7, Chapter 2] dy = yRγ (t)T dt

# of n ?rst order di?erential equations based on α ∈ Fn , where Rα (t) is de?ned as in Section 4

(see also [7, Chapter 2]). This is illustrated by the following example. Let n = 2, ? = 1. Take α = (u1 , u2 ) as in Section 3. Then the linear system based on α is ? ? ω1 (χ# (α)) dy1 dy2 t ?. ( , ) = (y1 , y2 ) ? dt dt ω2 (χ# ( α )) t We consider the case when limt→±∞ consider a perturbed system ( dy1 dy2 , ) = (y1 , y2 ) ? dt dt ? ω1 (χ# t (α)) ω2 (χ# t (α)) ? ?+? ? a a ? ?,

1 t t 0

# ωt (χ# t (α))dt = λ < 0 and ω2 (χt (α)) ≡ 0 ?t ∈ R. Let us

where a > 0 is a small constant. We get y2 (t) = at, and thus get 0 > lim which is a contradiction.

t→±∞

1 1 log (y1 (t), y2 (t) ≥ lim log |at| = 0, t → + ∞ t t

20

6

A persistence property for Liao perturbations

A nearby C 1 vector ?eld, while perturbing a given one, keeps neither value nor sign of Lyapunov exponents, in general. However, if we perturb the given C 1 vector ?eld by a “Liao perturbation”, we will show in this section that the perturbed vector ?eld will keep both sign and value of the nonzero Lyapunov exponents. The class of Liao perturbations is constructed using the standard system of the given vector ?eld. Recall that S is the C 1 vector ?eld on M n given in Section 1. It reduces then in Section 2 the ?ows φ : M n → M n ,

# # χ# : F n → Fn . Let ν ∈ E (M n , φ) denote the probability in the 1 4.

Main Theorem. Let η > 0 be small and let F (η ) be as in the Corollary 5.3, ? ?(F (η )) > Lemma 5.2, F (η ) ? id? (W ). Recall from Section 3 the projection map id? :

# Fn

From

→

# F? .

Now we recall brie?y the Liao standard system for a perturbation vector ?eld [7, Chapter 2] with respect to the orthonormal n-frame β we chose. Let us take and ?x x ∈ Λ and β ∈ W so that

# ? := id? (β ) ∈ F (η ). Construct a standard map Pβ : R × Rn → M n qn (β ) = x and β n

Pβ (t, y ) = exp(

i=1

y i proji χ# i (β )),

y = (y 1 , . . . , y n ).

As M n is a compact C ∞ Riemannian manifold, the exponential map exp : T M n → M n is C ∞ and there exists a constant ζ0 > 0 such that for any x ∈ M n , exp maps {u ∈ Tx M n | u < ζ0 } di?erentially into a neighborhood of x on M n . Let X be a C 1 vector ?eld, a perturbation to the given vector ?eld S . Fixing t ∈ R, there exists a unique tangent vector ?eld Xβ (t, y ) on B0 = {y ∈ Rn | y < ζ0 } so that dPβt (Xβ (t, y )) = dPβ (0, Xβ (t, y )) = X (Pβ (t, y )) ? dPβ ( The system dy = Xβ (t, y ) dt can be written as dy ?(t, y ), = yRβ (t)T + f dt (6.1) ? |(t,y) ). ?t

where Rβ (t)T = (rij )n×n is de?ned in Section 4, rii (t) = ωi (χ# t (β )), for i = 1, . . . , n. The vector ?(t, y ) is bounded and Lipschitz. The system (6.1), called the Liao standard system of function f

21

X based on (S, ν ), was employed by Liao to prove the C 1 closing lemma [7, Appendix A] and topological stability for Anosov ?ows [7, Chapter 2]. Based on the Liao standard system, we now introduce the terminology of Liao perturbation to the given vector ?eld (M n , S, ν ) in our Main Theorem. We de?ne a triangular ? × ? matrix A?×? (t) = ? (aij (t))?×? as follows: aij (t) = 0 if i < j ; aij (t) = r(n??+i)(n??+j ) (t) if i > j ; aii (t) = ω ? i (χ# t (β )), i, j = 1, . . . , ?. And de?ne a vector function f : R? → R by

1 ? ? fi (t, y ) = f n??+i (t, (0, . . . , 0(n ? ?), y , . . . , y ),

i = 1, . . . , ?.

We then call the system dy = yA?×? (t) + f (t, y ), dt the system (6.2) a Liao perturbation of (M n , S, ν ). From our Main Theorem we easily summarize the e?ect of Liao perturbations on nonzero Lyapunov exponents Theorem 6.1 Let S be a C 1 vector ?eld on M n and let ν ∈ E (M n , φ) be a probability that has ? nonzero Lyapunov exponents λ1 < . . . < λ? together with n ? ? zero Lyapunov exponents. Then there exists a C 1 neighborhood X 1 of S on the space of all C 1 vector ?elds on M n , so that for each X ∈ X 1 , its reduced standard system (6.2) based on (S, ν ) has λ1 , . . . , λ? as Lyapunov exponents. In other words, Liao perturbation preserves the nonzero Lyapunov exponents. Remark 6.2 Because the Lyapunov exponents are constant on Λ and F (η ) ? id? (W ), from Propo? ∈ F (η ) sition 3.4 and the Main Theorem, Theorem 6.1 is independent of the choice of x ∈ Λ and β and thus the reduced standard system based on (S, v ). (6.2)

a reduced standard system of the perturbation vector ?eld X based on (M n , S, ν ). Simply, we call

References

[1] J. F. Alves, SRB measures for nonhyperbolic systems with multidimensional expansions, preprint, IMPA, Brasil, 1998. [2] J. Bochi, Genericity of zero Lyapunov exponents, preprint, IMPA, Brazil, 2000. [3] H. Fedrer, Geometric measure theory, Springer-Verlag, 1969. [4] S. T. Liao, Certain ergodic property theorem for a di?erential systems on a compact di?erentiable manifold, Acta Scientiarum Naturalium Universitatis Pekinesis 9, 241–265, 309–327 (in Chinese) (1963). Its English version appears as Chapter 1 in the book of S. T. Liao, Qualitative theory on di?erentiable dynamical systems, Science Press, Beijing, New York, 1996. 22

[5] S. T. Liao, An ergodic property theorem for a di?erential system, Science in China 16, 1-24 (1973). [6] S. T. Liao, On characteristic exponents construction of a new Borel set for multiplicative ergodic theorem for vector ?elds, Acta Scientiarum Naturalium Universitatis Pekinesis 29, 177-302 (1992). [7] S. T. Liao, Qualitative theory on di?erentiable dynamical systems, Science Press, Beijing, New York, 1996 [8] S. T. Liao, Notes on a study of bundle dynamical systems(II), part 1, and part 2, Appl. Math. Mechanics (English Edition) 17, 805-818 (1996), 18, 421-440 (1997). [9] B. B. Nemytskii, B. B. Stepanov, Qualitative theory of di?erential equations, Princeton University Press, 1960 [10] V. I. Oseledec, A multiplicative ergodic theorem, Lyapunov characteristic number for dynamical systems, Trans. Moscow Math. Soci. 19, 197-231 (1968). [11] Jacob Palis, A global view of dynamics and conjecture on the denseness of ?nitude of attractors, Asterisque 261, 339-351 (1999). [12] W. Sun, Characteristic spectrum for di?erential systems, J. Di?. Equations 147, 184-194 (1998). [13] W. Sun, Qualitative functions and characteristic spectra for di?eomorphisms, Far East J. Appl. Math. 2, 169-182 (1998). [14] W. Sun, Characteristic spectra for paralellotope cocycles, in Dynamical Systems, World Scienti?c, Singapore, 256-265, 1999. [15] W. Sun, Entropy of orthonormal n-frame ?ows, Nonlinearity 14, 892-842 (2001). [16] M. Viana, Multidimensional nonhyperbolic attractors, Publ. Math. IHES 85, 63-96 (1997). [17] L. -S. Young, Some open sets of nonuniformly hyperbolic cocycles, Erg. Th. & Dynam. Sys. 13, 409-415 (1993). [18] L. -S. Young, Lyapunov exponents for some quasi-periodic cocycles, Erg. Th. & Dynam. Sys. 17, 483-501 (1997).

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