Approximation Properties on Invariant Measure and Oseledec Splitting in Non-Uniformly Hyperbolic Systems

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 3, March 2009, Pages 1543–1579 S 0002-9947(08)04630-8 Article electronically published on October 21, 2008

APPROXIMATION PROPERTIES ON INVARIANT MEASURE AND OSELEDEC SPLITTING IN NON-UNIFORMLY HYPERBOLIC SYSTEMS

CHAO LIANG, GENG LIU, AND WENXIANG SUN

Abstract. We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Deﬁnition 1.10), independence number (Deﬁnition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic periodic orbits, respectively. Combining this result with the approximation property of Lyapunov exponents by Wang and Sun, 2005 (Theorem 1.9), we strengthen Katok’s closing lemma (1980) by presenting more extensive information not only about the state system but also its linearization. In the present paper, we also study an ergodic theorem and a variational principle for mean angle, independence number and Liao’s style number (Def- inition 1.3) which are bases for discussing the approximation properties in the main result.

1. Introduction Let f : M → M be a C1 diﬀeomorphism of a compact smooth Riemannian manifold M and let Df : TM → TM be the tangent map on the tangent bundle ≤ ≤ U # TM . For any integer ,1 dimM = d, we construct a bundle (M)= U # x∈M (x)of-frames, where the ﬁber over x is U # { ∈ × ×···× | (x)= α =(u1,u2,...,u) TxM TxM TxM ui =1,i=1, ..., ,

and u1,u2,...,u are linearly independent vectors}. V# V# Set (M)= x∈M (x), where the ﬁber over x is V# { ∈ × ×···× | (x)= α =(u1,u2,...,u) TxM TxM TxM ui =1,i=1, ..., ,

and u1,u2,...,u are linearly dependent vectors}. # U # ∪V# # Let L (M):= (M) (M); then L (M) forms a compact metric space. Let # → π : L (M) M be the natural bundle projection. The tangent map Df induces

Received by the editors March 2, 2007. 2000 Mathematics Subject Classiﬁcation. Primary 37C40, 37D25, 37H15, 37A35. Key words and phrases. Independence number, invariant measure, mean angle. The ﬁrst and second authors were supported by NNSFC(# 10671006). The third author was supported by NNSFC (# 10231020, # 10671006), the National Ba- sic Research Program of China (973 Program)(# 2006CB805900) and the Doctoral Education Foundation of China.

c 2008 American Mathematical Society Reverts to public domain 28 years from publication 1543

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# # amapD f on L (M): # # → # D f : L (M) L (M),

# Df(u1) Df(u2) Df(u) D f(u1,u2,...,u)=( , ,..., ). Df(u1) Df(u2) Df(u) It is clear that π ◦ D#f = f ◦ π. − ∈U# ∈ For an frame α =(u1,u2, ..., u) (x)atapointx M, we define Vol(α)=Vol(u1,u2, ..., u) as the volume of the parallelepiped generated by the vectors u1,u2, ..., u. More precisely, we choose an orthonormal −frame β = (w1,w2, ..., w),wi ∈ TxM, i =1, ..., , which generates the same linear subspace of TxM as α does, and we take a unique × matrix A with α = βA. Then we define the volume of α by Vol(α):=|det A|. We remark that the volume Vol(α) does not depend on the choice of β,sincethe determinate of a transition matrix between two orthonormal frames is ±1. Hence, Vol(α) is well defined. ∈U# ∈ Definition 1.1 ([6]). Aframeα =(u1,u2, ..., u) (x),x M, is called positive mean linearly independent if there is >0 such that the characteristic function χα on the set {n ∈ Z+ | Vol(D#f n(α)) ≥ } has positive time mean; that is,

1 n−1 lim → ∞ Σ χ (i) > 0. n + n i=0 α If for any positive number , the above supper limit is zero, α is called positive mean # i linearly dependent. Sometimes we write χα(i)asχα(D f ) to emphasize the i−th iteration of D#f. Similarly, one can deﬁne negative mean linear independence and dependence by using the negative time mean

1 n−1 # i lim →−∞ Σ χ (D f ). n |n| i=0 α ∈U# · · Let α =(u1, ..., u) . Denote by A(α)thematrix(ui,uj )×, where , is the inner product induced by the Riemannian metric. Let σ(α)denotetheset of all eigenvalues of A(α)andletτ(α) be the smallest eigenvalue. Note that A(α) is a real positive-definite symmetric matrix, therefore, σ(α) ⊂ (0, +∞). Now we give an alternative description of mean linear independence considering the average change of τ(α) over time instead of that of Vol(α) (for the equivalence of these two definitions see Proposition 2.3). ∈U# Definition 1.2. Aframeα is positive mean linearly independent if and only if 1 n−1 # i lim → ∞ Σ τ(D f α) > 0. n + n i=0 It is negative mean linearly independent if and only if

1 n−1 # i lim →−∞ Σ τ(D f α) > 0. n |n| i=0

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Let X be a metric space. We denote by Cb(X, R) the set of all bounded continuous functions on X. Given a continuous map g : X → X,wedenoteby Merg(X, g)thesetofallg−invariant and ergodic measures. Then we deﬁne for m ∈Merg(X, g), { ∈ | 1 n−1 i 1 n−1 i Qm(X, g):= x X lim Σi=0 φ(g (x)) = lim Σi=0 φ(g (x)) n→+∞ n n→−∞ n = φdm, ∀φ ∈ Cb(X, R)}. X

We call Qm(X, g) the basin of m. By the Birkhoff Ergodic Theorem Qm(X, g)is b an m−full measure set and g(Qm(X, g)) = Qm(X, g). We can replace C (X, R)by 0 C (X, R), the set of all continuous functions on X,inQm when X is compact. Definition 1.3. (1) Let x ∈ M and set ∗ { |∃ ∈U# } k+(x):=max α (x), s.t.α is positive mean linearly independent and ∗ { |∃ ∈U# } k−(x):=max α (x), s.t.α is negative mean linearly independent . ∗ ∗ We call k+(x)andk−(x) the positive and the negative style number of f at x, respectively. ⊂ − ∗ (2) Let F M be an f invariant subset. The positive style number k+(F )and ∗ thenegativestylenumberk−(F )ofF are defined as ∗ ∗ ∗ ∗ k+(F ):=supk+(x)andk−(F ):=supk−(x), x∈F x∈F respectively. The style number k∗(F )ofF is defined by ∗ { ∗ ∗ } k (F ):=max k+(F ),k−(F ) .

(3) Let m ∈Merg(M,f). The positive style number, the negative style number and the style number of m are given by, respectively, ∗ ∗ ∗ ∗ k+(m):=k+(Qm(M,f)),k−(m):=k−(Qm(M,f)) and ∗ { ∗ ∗ } k (m):=max k+(m),k−(m) . Definition 1.3 (1)(2) are the diffeomorphism version of Liao’s style number in [6]. Definition 1.3 (3), adapted by Dai [2], is a special case of Definition 1.3 (2) with F = Qm(M,f). One may replace Qm(M,f) by any invariant subset of m−total measure, for instance, Qm(M,f) ∩ supp(m), where supp(m) denotes the support of m. The following is the diffeomorphism version of the main theorem concerning style number in [6]. Theorem 1.4 ([6]). Let f : M → M be a C1 diffeomorphism of a compact smooth Riemannian manifold and let F ⊂ M be a closed f−invariant set. Then ∗ ∗ ∗ (1) k−(F )=k+(F )=k (F ) k. ∈ ∈U# − (2) There is x M and α k (x) such that the orbit Orb(x) is f recurrent and

1 n−1 # i 1 n−1 # i lim limn→+∞ Σ χα(D f ) = lim limn→−∞ Σ χα(D f )=1. →0 n i=0 →0 |n| i=0

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One result in the present paper is an ergodic theorem (Theorem 1.5) about mean linear independence and style number. Note that the basin of a given ergodic measure is contained in the Birkhoﬀ center ([8], [19] and [20]) and thus always contains recurrent orbits. Hence Theorem 1.5 is a generalization of Theorem 1.4. Moreover, the upper limit in Theorem 1.4 could be exactly the limit. Theorem 1.5. Let f : M → M be a C1 diﬀeomorphism of a compact smooth Riemannian manifold. Let m ∈Merg(M,f). Then (1) ∗ ∗ ∗ ∗ k−(x)=k−(m),k+(x)=k+(m), for m − a.e. x ∈ M and ∗ ∗ ∗ k−(m)=k+(m)=k (m) k. − ∈ − ∈U# (2) For m a.e. x M there is an orthonormal k frame α k (x) such that

1 n−1 # i 1 n−1 # i (1.1) lim lim Σ χα(D f ) = lim lim Σ χα(D f )=1. →0 n→+∞ n i=0 →0 n→−∞ |n| i=0 ∈M U # # ∈U# (3) There exists µ erg( k ,D f),π∗(µ)=m, such that the set of all α k satisfying (1.1) is µ−full measure. ∈U# Deﬁnition 1.6. (1) Let α . We call the supper limit

1 n−1 # i τ˜(α):=lim → ∞ Σ τ(D f α) n + n i=0 the independence number for α. M U # # ∅ ∈M U # # (2) If erg( ,D f) = , we define for µ erg( ,D f) { | ∈ U # # } τ˜(µ):=sup τ˜(α) α Qµ( ,D f) and callτ ˜(µ)theindependence number for µ. ∈V# · # R For α , we defineτ ˜(α)=0.Then˜τ( ) is a function from L to . Remark. Replacing τ(D#f iα)byVol(D#f iα) in Definition 1.6 one gets an alternative description of independence numbers (ref. Proposition 2.3). The next theorem is a variational principle for style numbers and independence numbers, which asserts that the biggest style number can be achieved at an ergodic measure (on the state manifold) and at the same time, and that the largest independence number can be achieved at an ergodic measure (on the unit frame bundle) covering the one with the biggest style number. Theorem 1.7. Let f : M → M be a C1 diffeomorphism on a compact smooth ∗ Riemannian manifold and let k = k (M). Then there exist m0 ∈Merg(M,f) and ∈M U # # ν0 erg( k ,D f), with π∗(ν0)=m0, such that ∗ ∗ ∗ ∗ k (x)=k (m0)=max{k (m) | m ∈Merg(M, f)} = k (M),m0 − a. a. x ∈ M and { | ∈M U # # } − ∈U# τ˜(β)=˜τ(ν0)=sup τ˜(µ) µ erg( k ,D f) =sup τ˜(α),ν0 a. a. β k . ∈ # α Lk (M)

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We will present some examples in Section 2.2 to compare the style number with the number of Lyapunov exponents and the dimension of the state manifold. In Sections 2.3 and 2.4 we will prove Theorem 1.5 and Theorem 1.7, respectively. In [2] Dai obtained certain results concerning the style number in the direction of Theorem 1.5 and Theorem 1.7 for C1 vector fields. Theorem 1.5 and Theorem 1.7 will contribute to the proof of our main result, Theorem 1.12. Another crucial result in its proof is the following theorem on measure approximation. We denote by MHP (f) the set of atomic measures on hyperbolic periodic orbits. Theorem 1.8. Let f : M → M be a C1+α(α>0) diffeomorphism on a compact manifold. Suppose that f preserves an ergodic hyperbolic measure ω.˜ Then there exists an f−invariant set Λ˜ of ω˜−full measure such that MHP (f) is dense in Minv(Λ)˜ ,thesetofallf−invariant measures supported by Λ˜. Theorem 1.8 is a generalization of Sigmund’s result for uniform hyperbolicity [14] and Hirayama’s result for mixing non-uniform hyperbolicity [3]. We will prove Theorem 1.8 in Section 3. Let us recall the Oseledec theorem and give some necessary notions. The Oseledec Theorem ([9]). Let f be a C1 diffeomorphism of a compact d- dimensional Riemannian manifold M preserving an ergodic measure ω˜. Then there exist

(a) real numbers λ1 < ···<λs(s ≤ d); (b) positive integers n1,...,ns, satisfying n1 + ···+ ns = d; (c) a Borel set O(˜ω), called an Oseledec basin of ω˜, satisfying f(O(˜ω)) = O(˜ω) and ω˜(O(˜ω)) = 1; (d) a measurable splitting, called an Oseledec splitting, TxM = E1(x) ⊕···⊕ Es(x) with dimEi(x)=ni and Df(Ei(x)) = Ei(fx), such that log Dfnv lim = λi, n→±∞ n for all x ∈ O(˜ω), v ∈ Ei(x),i=1, 2, ..., s. (a) and (d) in the Oseledec Theorem allow us to arrange the Oseledec splitting according to the increasing order of the Lyapunov exponents. To avoid excessive terminology, we will arrange the Oseledec splitting at every point in the Oseledec basin in this way throughout this paper without explanation. We call the measure ω˜ hyperbolic if none of its Lyapunov exponents is zero. The following theorem is the main theorem in [18]. Theorem 1.9 ([18]). Let M be a compact d−dimensional Riemannian manifold. Let f : M → M be a C1+α diffeomorphism, and let ω˜ be an ergodic hyperbolic measure with Lyapunov exponents λ1 ≤ ···≤λr < 0 <λr+1 ≤ ···≤λd. Then the Lyapunov exponents of ω˜ can be approximated by Lyapunov exponents of hyperbolic periodic orbits. More precisely, for any >0, there exists a hyperbolic periodic z ≤···≤ z | − z | point z with Lyapunov exponents λ1 λd such that λi λi <,i=1, ..., d. Definition 1.10. Let E and F be two Df−invariant sub-bundles of TM.The angle between E(x)andF (x), x ∈ M,isdefinedasfollows: u ∧ v sin ∠(E(x),F(x)) := inf sin ∠(u, v)= inf , 0= u∈E(x),0= v∈F (x) 0= u∈E(x),0= v∈F (x) uv

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where ∧ denotes the wedge product. We call − 1 n1 m∠(E(x),F(x)) := lim ∠(DfiE(x),DfiF (x)) n→+∞ n i=0 the mean angle between E and F at x. Suppose that f preserves an ergodic measureω ˜ with an Oseledec splitting

TxM = E1(x) ⊕···⊕Es(x),s≤ d = dimM, x ∈ O(˜ω). By the Birkhoﬀ Ergodic Theorem there is anω ˜−full measure subset in O(˜ω) such that every point x in this subset satisﬁes

m∠(Ei(x),Ej(x)) = ∠(Ei(y),Ej(y))dω˜(y). ∠ We call (Ei(y),Ej(y))dω˜(y) the mean angle between Ei := x∈O(˜ω) Ei(x)and ∠ ∀ ≤ ≤ Ej := x∈O(˜ω) Ej(x)andwriteitasm ω˜ (Ei,Ej), 1 i = j d. If, in addition,ω ˜ has d diﬀerent Lyapunov exponents, namelyω ˜ has a simple spectrum, the Oseledec splitting will be

(1.2) TxM = E1(x) ⊕···⊕Ed(x),x∈ O(˜ω). For x ∈ O(˜ω), we define the independence number of x by the independence number of frames at x whose elements are all on different invariant bundles. More precisely, ∈U# ∈ take α =(u1,u2, ..., ud) d ,whereui Ei(x),i=1, 2, ..., d. Then we define τ(x):=τ(α), the smallest eigenvalue of A(α). Clearly, τ(x) is well defined for x ∈ O(˜ω). Moreover, by the Birkhoff Ergodic Theorem, the equation − 1 n1 lim τ(f ix)= τ(y)dω˜(y) n→+∞ n i=0 holds on anω ˜−full measure subset of O(˜ω). Therefore, we can define the independence number ofω ˜ by τ˜(˜ω):= τ(y)dω˜(y).

Assume there is another ergodic hyperbolic measureω ˜ with simple spectrum. We split the tangent bundle TM into invariant bundles (1.3) TyM = E1(y) ⊕ E2(y) ⊕···⊕Ed(y),y∈ O(˜ω ). Additionally, we assume thatω ˜ andω ˜ have the same number of negative Lyapunov exponents (and thus the same number of positive ones). Under these assumptions we further describe the approximation of Oseledec splittings. Remember both (1.2) and (1.3) are arranged according to the increasing order of Lyapunov exponents. Deﬁnition 1.11. Let η>0. The Oseledec splitting (1.2) ofω ˜ is η approximated by (1.3) ofω ˜ if there exists a measurable subset A satisfying: (1)ω ˜(A) > 1 − η;

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∈ ∈U# ∈ (2) for any x A and any frame α =(u1,...,ud) d (x)withui Eσ(i)(O(˜ω)), there exist a point z = z(x) ∈ Supp(˜ω ) ∩ O(˜ω )andaframe ∈U# ∈ β =(v1,...,vd) d (z)withvi Eσ(i)(O(˜ω )), such that dist(α, β) <η, where σ : {1,...,d}→{1,...,d} is a permutation, and dist denotes the metric on the Grassman bundle. This deﬁnition does not depend on the choice of α or β. Now we state our main result of this paper. Theorem 1.12. Let f : M → M be a C1+α diﬀeomorphism preserving an ergodic hyperbolic measure ω˜ which has a simple spectrum and denote d = dimM. Let

TΛM = E1(Λ) ⊕ E2(Λ) ⊕···⊕Ed(Λ) be the Oseledec splitting of ω˜arranged according to the increasing order of Lyapunov exponents of ω˜,whereΛ= k≥1 Λk is the Pesin set associated with ω˜(see Section 3 for a deﬁnition). Given ε>0, there is a hyperbolic periodic orbit Orb(z, f) together with its Oseledec splitting

TzM = E1(z) ⊕ E2(z) ⊕···⊕Ed(z) at z arranged according to the increasing order of Lyapunov exponents of the orbit Orb(z) such that the atomic measure ωz supported on Orb(z,f) satisﬁes the following properties:

(1) Mean angles of ω˜ and of ωz are ε−close; that is, | ∠ − ∠ | ∀ ≤ ≤ m ω˜ (Ei(Λ),Ej(Λ)) m ωz (Ei(Orb(z)),Ej(Orb(z))) <ε, 1 i = j d.

(2) Independence numbers of ω˜ and of ωz are ε−close; that is,

|τ˜(˜ω) − τ˜(ωz)| <ε.

(3) The Oseledec splitting of ω˜ is ε−approximated by that of ωz. Theorem 1.9 implies that ifω ˜ has a simple spectrum, so do the atomic measures on the hyperbolic periodic orbits chosen in the theorem. It motivates us to investigate more approximation properties of mean angle, independence number and Oseledec splitting than we listed in Theorem 1.12. Observe that the Oseledec splitting is continuous neither with state points x ∈ M nor with measures, and thus Theorem 1.9 cannot imply the approximation properties mentioned. A new approach would be required to prove Theorem 1.12. Here is how the proof goes. Let f : M → M be a C1+α diffeomorphism on a compact manifold of dimension d, preserving an ergodic hyperbolic measureω ˜ with a simple spectrum. We first choose a sequence of hyperbolic periodic orbits so that the i−th biggest exponent of every periodic orbit is close to the i−th biggest one ofω ˜,1≤ i ≤ d (see Theorem 1.9), and moreover, the atomic measures supported ontheseperiodicorbitsconvergeto˜ω (see Theorem 1.8). This is a crucial step in the proof since therefore we can transfer the properties ofω ˜ onto periodic orbits which are more flexible thanω ˜ itself. For each atomic measure, we can construct U # an ergodic measure on k to cover it, where k is the style number. This is from Theorem 2.7. Then we get a collection of ergodic measures on the bundle. Each U # of the limit measures of this collection coversω ˜.Notethat k is not compact, U # and hence the case that no limit measure is supported on k may happen. The

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key observation that helps us to overcome this obstacle is that a limit measure U # can be decomposed into the sum of finitely many ergodic measures on k covering ω˜ due to Sun and Vargas in [17]. Moreover, we show that not more than 2d of these have effects and are of the same type; that is, in every direction the integral Lyapunov exponents with respect to all these measures are equal (see Definition 4.2). That two ergodic measures are of the same type implies that they play the same role in discussing a mean angle ofω ˜ between two given directions. Thus we d U # focus on these not more than 2 measures on k coveringω ˜ and the mean angles of their generic frames. In this way we complete the verification of the approximation property about a mean angle. This analysis also helps us prove other approximation properties. Finally we summarize these results to obtain an extensive version of Katok’s closing lemma. Theorem 1.13. Let f : M → M be a C1+α diffeomorphism preserving an ergodic hyperbolic measure ω˜ and denote d = dimM. Let

TΛM = E1(Λ) ⊕ E2(Λ) ⊕···⊕Es(Λ),s≤ d, be the Oseledec splitting arranged according to the increasing order of Lyapunov exponents of ω˜,whereΛ= k≥1 Λk is the Pesin set associated with ω˜. Given ε>0, (a) for any integer k>0,thereisanumberβ = β(k, ε) > 0 with the property p p that if x, f (x) ∈ Λk, and d(x, f x) <βfor some positive integer p,then there exists a unique hyperbolic periodic point z ∈ M with period p, such that d(z, x) <ε;and (b) there is an Oseledec splitting TzM = E1(z) ⊕ E2(z) ⊕···⊕Es(z) at z,such that the Lyapunov exponents of z on Ei(z) are ε−close to the Lyapunov exponents of ω˜ on Ei(Λ) and dimEi(z)=dimEi(Λ), ∀ i =1, 2, ..., s;and 1 p (c) let ωz := p Σi=1δf iz denote the atomic measure on the periodic orbit orb(z). ∗ The distance between ω˜ and ωz is less than ε in weak topology; If in addition ω˜ has a simple Lyapunov spectrum, the following holds: (d) mean angles of ω˜ and of ωz are ε−close; that is, | ∠ − ∠ | ∀ ≤ ≤ m ω˜ (Ei(Λ),Ej(Λ)) m ωz (Ei(Orb(z)),Ej(Orb(z))) <ε, 1 i = j d;

(e) independence numbers of ω˜ and of ωz are ε−close; that is,

|τ˜(˜ω) − τ˜(ωz)| <ε;

(f) the Oseledec splittings of ω˜ are ε−approximated by that of ωz. Theorem 1.13 includes broader information than the original closing lemma of Katok [4], since it discusses not only the approximation property on the manifold but also the interrelated properties on its tangent bundles.

2. Style number and independence number 2.1. Mean linear independence. We will prove the equivalence of Deﬁnition 1.1 and Deﬁnition 1.2 in this subsection. ∈U# Lemma 2.1. Let α ;then

detA(α) 1 1 ≤ τ(α) ≤ (detA(α)) ≤ . −1

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{ } ≥ ≥ ··· ≥ Proof. Let σ(α)= τi i=1 with τ1 τ2 τ = τ(α). Since detA(α)= 1 ≥ ≤ Πi=1τi (τ(α)) , then τ(α) (detA(α)) . Write A(α)as(aij)× and let A = | | ≤ ∈ max1≤i≤ Σj=1 aij . Then A(α) . For each eigenvalue τi σ(α), take an eigenvector u with u =1. Then

τi = τiu = A(α)u≤A(α)u = A(α)≤. So we have −1 ≤ −1 det A(α)=Πi=1τi = τ(α)Πi=1 τi τ(α) and thus detA(α) 1 1 ≤ τ(α) ≤ (detA(α)) ≤ . −1

The proof of the following lemma is straightforward. ∈ ∈U# Lemma 2.2. Let x M and α =(u1, ..., u), β =(v1, ..., v) (x), where β is an orthonormal frame. Suppose that α and β generate the same subspace of TxM and α = βA;thendetA(α)=(detA)2 = Vol(α)2. Proposition 2.3. Definition 1.1 and Definition 1.2 are equivalent. ∈U# Proof. Suppose that a frame α satisfies

1 n−1 # i τ˜(α)=lim → ∞ Σ τ(D f α) > 0. n + n i=0 { }+∞ ≥ Then there exist a subsequence nj j=1 and a real 1 δ>0 such that

1 nj −1 # i lim Σi=0 τ(D f α)=2δ. j→+∞ nj For any 0 ≤ µ ≤ 1, set # i N(µ):=#{0 ≤ i ≤ nj | τ(D f α) ≥ µδ}. # i 1 Note that τ(D f α) ≤ ≤ 2 by Lemma 2.1, so we have (n − N(µ))µδ +2N(µ) j >δ nj − as j is large enough. Let ι(µ):= N(µ) ;thenι(µ) > δ µδ . Specially, for µ = 1 , it nj 2−µδ 2 1 δ 1 2 holds that ι( 2 ) > 4−δ . Take := ( 2 δ) ; then by Lemma 2.1 and Lemma 2.2, we have # i #{0 ≤ i ≤ nj | Vol(D f α) ≥ } 1 ≥#{0 ≤ i ≤ n | τ(D#f iα) ≥ δ} j 2 1 =N( ). 2 Thus 1 1 − N( ) 1 δ nj 1 # i ≥ 2 Σi=0 χα(D f ) = ι( ) > , nj nj 2 4 − δ and hence, 1 n−1 # i lim → ∞ Σ χ (D f ) > 0. n + n i=0 α So Deﬁnition 1.2 implies Deﬁnition 1.1.

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∈U# On the contrary, for a frame α with

1 n−1 # i lim → ∞ Σ χ (D f ) > 0, n + n i=0 α { }+∞ ≥ there exist a subsequence nj j=1 and a real 1 δ>0 such that

1 nj −1 # i lim Σi=0 χα(D f )=2δ. j→+∞ nj By Lemma 2.1 and Lemma 2.2, we have 1 2 #{0 ≤ i ≤ n | τ(D#f iα) ≥ }≥#{0 ≤ i ≤ n | Vol(D#f iα) ≥ }. j −1 j So 1 1 − 2 nj 1 # i ≥ Σi=0 τ(D f α) −1 δ>0 nj as j is large enough. Hence,

1 n−1 # i lim →∞ Σ τ(D f α) > 0. n n i=0 Thus Definition 1.1 implies Definition 1.2. 2.2. Style number, number of Lyapunov exponents and dimension of state manifold. We will compare the style number of a diffeomorphism with the number of its Lyapunov exponents and the dimension of the state manifold in this subsection. There are many cases where the style number coincides with the number of Lyapunov exponents, for instance, the well-known Thom automorphism on T2.In the following we present our first example which shows that the two numbers are different. Example 1. Consider a C1 diffeomorphism f on a 2−dimensional manifold M n−1 n and a periodic orbit orb(x, f)={x, f(x), ..., f (x)} satisfying Dxf =3I2×2 : TxM → TxM, where I2×2 denotes the 2 × 2 unit matrix. Let m be the atomic measure on this orbit. Then log 3 is the unique Lyapunov exponent. Take and fix an # n orthonormal frame α =(e1(x),e2(x)) in TxM. Since Vol(D f (e1(x),e2(x))) = 1 ∀n ∈ Z+, then k∗(m)=2. The style number of a diffeomorphism is not less than the number of its Lyapunov exponents, since we can construct a mean linearly independent frame each of whose vectors is contained in some invariant bundle by the Oseledec Theorem. Our next example, an adaption of Liao’s example in [6], shows that the style number may be strictly smaller than the dimension of M. Example 2. Let φ : R → R be a C∞ bump function which satisfies 1 3 φ(x)=0 for x ≤ or x ≥ 2 2 and dφ | = b =0 . dx x=1 Define ψ : R2 → R2,(r, θ) → (r, θ + φ(r)), in polar coordinates. Extend ψ to the sphere S2 = R2 ∪{∞}and thus obtain a new diffeomorphism f : S2 → S2, with f(z)=ψ(z),z∈ R2. Then Γ : r =1isanf-invariant closed set in the manifold

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S2 ∈ ,andforanypointz Γ and any orthonormal frame αz =(uz,uz)atz,itholds that

n Dzf (αz) =(uf n(z),nbuf n(z) + uf n(z)) n =(uf n(z),uf n(z))B n =αf n(z)B , where 1 b B = . 01 ∈U# Atthesametimewecanﬁndforanyγz =(wz,wz) 2 (z) a non-degenerate matrix c1 c1 C = c2 c2

such that γz = αzC. Thus n n Dzf (γz)=αf n(z)B C =((c1+nbc2)uf n(z)+c2uf n(z), (c1+nbc2)uf n(z)+c2uf n(z)). Denote wzn =(c1 + nbc2)uf n(z) + c2uf n(z), wzn =(c1 + nbc2)uf n(z) + c2uf n(z). Therefore, n | n|| | | | # n Vol(Df (γz)) Vol(αf n(z)) detB detC detC Vol(D f (γz)) = = = . wzn wzn wzn wzn wzn wzn Since detC =0,c2 and c2 cannot vanish simultaneously. This means either →∞ →∞ → ∞ # n → wzn or wzn as n + . Consequently, Vol(D f (γz)) 0as ∗ 2 n →∞, and then by the arbitrariness of z and γz,wehavek (Γ) = 1 < 2=dimS . By choosing a suitable function φ : R → R, one constructs a diﬀeomorphism f and a periodic orbit for f on Γ. Then the atomic measure on this periodic orbit 2 m ∈Merg(S ,f)satisﬁes k∗(m)=1

Let m ∈Merg(M,f). Theorem 1.5 states that for m − a.e. x ∈ M there exists ∗ ∈U# a mean linearly independent k (m)-frame α k∗(m)(x), and all these frames in U # − ∈M U # k∗(m) form a µ full measure subset for some µ erg( k∗(m))withπ∗(µ)=m. ∗ ∈U# The next example shows that there possibly exists a k (m)-frame β k∗(m)(x) that is not mean linearly independent, even if it generates the same linear subspace in TxM as a mean linearly independent frame generates. Example 3. Let m be an ergodic measure on M with k∗(m)=k(m)=dimM =2 and let λ1 >λ2 be its Lyapunov exponents with diﬀerent signs. Take for m − a.e. x ∈ M the invariant and orthogonal splitting TxM = E1(x) ⊕ E2(x)asin the Oseledec Theorem [9]. Take and ﬁx unit vectors e1 in E1(x)ande2 in E2(x).

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Then α := (e1,e2) is mean linearly independent by Deﬁnition 1.1 and the Birkhoﬀ Ergodic Theorem. Let e + e e − e β := ( 1 2 , 1 2 ). e1 + e2 e1 − e2 Then n # n Vol(Df β) Vol(D f β)= − . Dfn e1+e2 Dfn e1 e2 e1+e2 e1−e2

Since neither e1 + e2 nor e1 − e2 is contained in any invariant bundle, we have 1 e ± e lim log Dfn 1 2 = λ . → ∞ 1 n + n e1 ± e2 On the other hand, we deduce from [15], [16] that

1 n 1 n lim log Vol(Df β) = lim log Vol(Df α)=λ1 + λ2. n→+∞ n n→+∞ n Thus 1 # n lim log Vol(D f β)=λ2 − λ1 < 0, n→+∞ n and hence Vol(D#f n(β)) → 0asn → +∞, which implies that β is not mean linearly independent by Definition 1.1. 2.3. A continuity property for τ and τ˜. In this subsection we will prove the con- U # → ∞ M U # # → tinuity of τ : (0, + ) given before Definition 1.2 andτ ˜ : erg( ,D f) [0, +∞) given in Definition 1.6, which are necessary for proving Theorem 1.5 and Theorem 1.7. Denote by L(R) the space of all linear maps on R and by σ(L)thesetofall eigenvalues of a linear map L ∈L(R). For any >0, set σ(L):={τ ∈ C |∃ τ ∈ σ(L),s.t. |τ − τ | <}. Lemma 2.4 and Lemma 2.5 are cited from [10].

Lemma 2.4. For given L ∈L(R ) and >0 there is δ>0 such that σ(T ) ⊂ σ(L) whenever T − L <δ. Lemma 2.5. Let L ∈L(R) and τ ∈ σ(L). There exist >0 and δ>0 with the property that for any T ∈L(R ) with T − L <δthe cardinality of σ(T ) ∩ B(τ) is not more than the multiplicity of τ, where B(τ)={τ ∈ C ||τ − τ | <}. U # → ∞ M U # # → Proposition 2.6. The function τ : (0, + ) and τ˜ : erg( ,D f) [0, +∞) are continuous. Proof. Since A(α) is continuous with α, we can deﬁne d(α, β):=A(α) − A(β) ∈U# when α, β are close enough. Assume that τ is not continuous at a frame ∈U# α0 . Then there must exist a constant 0 > 0 such that for any δ>0wecan ∈U# | − | ﬁnd another frame β = β(δ) satisfying d(α0,β) <δbut τ(α0) τ(β) > 20. Arrange the eigenvalues of A(α0) in decreasing order as {τ1 >τ2 > ··· >τt > τ(α0)},t≤ − 1. By Lemma 2.5, there are real numbers i > 0,δ1 > 0andδ > 0

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∈U# such that for any γ with d(γ, α0) <δ, the following holds: ∩ (a) the cardinality #σ(γ) Bi (τi) is not more than the multiplicity of τi, and ∩ (b) the cardinality #σ(γ) Bδ1 (τ(α0)) is not more than the multiplicity of τ(α0)and ∩ ∅ (c) Bi (τi) Bδ1 (τ(α0)) = ,i=1, ..., t. Take

0 <≤ min{i,δ1; i =0, 1, ..., t}. { } ∈U# Choose δ

< #σ(α0)=dimM. It’s a contradiction. This implies that the function τ is continuous. Additionally, τ is a bounded function by Lemma 2.1. ∈M U # # Let µ erg( ,D f). By the Birkhoff Ergodic Theorem, we have − 1 n1 τ˜(µ) = lim τ(D#f iα)= τdµ n→+∞ n i=0 ∈ U # # for any α Qµ( ,D f). Combining this equation with the continuity of τ,one can deduce thatτ ˜ is also continuous in the weak* topology. Hence the proposition is proved. 2.4. ProofofTheorem1.5. In this subsection we will choose an ergodic measure U # on k ,wherek is the style number, to cover the given ergodic measure on the state manifold and then prove Theorem 1.5. Theorem 2.7. Let f : M → M be a C1 diffeomorphism preserving an ergodic mea- ∈M ∗ ∗ ∗ sure m erg(M,f). Denote by k the style number k+(m),k−(m) or k (m), re- U # U # spectively. Denote by k (Qm(M,f)) the restriction bundle of k (M) on Qm(M,f). # − ∈M U # Then there exists a D f invariant ergodic measure µ erg( k (Qm(M, f))) so that π∗(µ)=m. Conversely, if a positive integer satisfies −1 ∩M U # # ∅ π∗ (m) erg( ,D f) = , it holds that ≤ k. ∗ Proof. We prove the case when k = k+(m) only, and omit the similar proof for the ∗ ∗ other cases of k−(m)andk (m). Denote by F := Closure(Qm(M, f)) the closure # # of Qm(M, f); then F is an f-invariant closed subset of M. Note (Lk (F ),D f)isa M # # continuous homeomorphism on a compact metric space and thus erg(Lk ,D f) = ∅ ◦ # ◦ ∀ ∈ # . Clearly, π D f(α)=f π(α), α Lk (F ).

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# # Step I. Find all ergodic measures on (Lk (F ),D f)coveringm. ∈ ∈U# Take and ﬁx a point x Qm(M, f), a frame α k (x) and a positive real >0 so that

1 n−1 (2.1) lim → ∞ Σ χ (i) > 0. n + n i=0 α # Deﬁne a sequence of measures µn on L by k 1 φdµ := Σn−1φ(D#f iα), ∀φ ∈ C0(L#, R). n n i=0 k

By taking a subsequence when necessary we can assume that µn → µ0. It is standard # to verify that µ0 is a D f-invariant measure and µ0 covers m, i.e., π∗(µ0)=m. Set # # # # Q(Lk ,D f):= Qν (Lk ,D f). ∈M # # ν erg (Lk ,D f) # # # − # Then Q(Lk ,D f)isaD f invariant total measure subset in Lk . We have ∩ # # m(Qm(M,f) πQ(Lk ,D f)) ≥ −1 ∩ # # µ0(π Qm(M,f) Q(Lk ,D f)) =1. Then the set A { ∈M # # |∃ ∈ # # ∈ := µ erg(Lk ,D f) γ Q(Lk ,D f),π(γ) Qm(M,f),s.t.

1 n−1 # i 1 n−1 # i lim Σi=0 φ(D f γ) = lim Σi=0 φ(D f γ) n→+∞ n n→−∞ n = φdµ, ∀φ ∈ C0(L#, R)} # k Lk

is non-empty. It is clear that µ covers m, π∗(µ)=m, for all µ ∈A. Also, we A M # # claim that coincides with the set of all the measures in erg(Lk ,D f)that ∈M # # cover m. In fact, if µ erg(Lk ,D f)coversm, π∗µ = m, from the fact that # # µ(Qµ(Lk ,D f)) = 1, we have ∩ # # m(Qm(M,f) πQµ(Lk ,D f)) ≥ −1 ∩ # # µ(π Qm(M,f) Qµ(Lk ,D f)) =1.

∈ # # ∈ ∈A Thus there is β Qµ(Lk ,D f)withπ(β) Qm(M,f), which means µ . Therefore, A { ∈M # # | } = µ erg(Lk ,D f) π∗(µ)=m . ∈A # # ⊂ Step II. Choose for m acoveringmeasureµ so that Qµ(Lk ,D f) U # k (Qm(M,f)). ∈M # # \A When µ erg(Lk ,D f) , then π∗(µ) = m by Step I, and thus −1 ∩ # # ∅ π Qm(M,f) Qµ(Lk ,D f)= .

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For the reference measure µ0 deﬁned in Step I, it follows that # # ∩ µ0( (Qµ(Lk ,D f) Fµ) ∈A µ −1 ∩ # # ∩ = µ0(π Qm(M,f) ( (Qµ(Lk ,D f) Fµ))) ∈A (2.2) µ −1 ∩ # # ∩ = µ0(π Qm(M,f) ( (Qµ(Lk ,D f) Fµ))) ∈M # # µ erg (Lk ,D f) =1

where Fµ is a µ−full measure set, µ ∈A. We claim that U # ∩ # # ∩ (2.3) µ0( k (F ) ( (Qµ(Lk ,D f) Fµ))) > 0 µ∈A

for any µ−full measure set Fµ,whereµ ∈A. ∈U# ∈ Let us recall the chosen frame α k (x), where x Qm(M,f), and the chosen # → R ≤ ≤ real >0 as in (2.1). We deﬁne a continuous function φ : Lk with 0 φ 1 { ≤ 3 } { ≤ ≤ } so that φ β : Vol(β) 4 =0andφ β : Vol(β) 1 = 1 by Urysohn’s # Lemma. Denote by χ{ ≤ ≤ } the character function on {β ∈ L : ≤ β: 2 Vol(β) 1 k 2 Vol(β) ≤ 1}. Then φ ≤ χ{ ≤ ≤ }. β: 2 Vol(β) 1

Note that the sequence of measures {µn} deﬁned below (2.1) converge to µ0.It holds that µn{β : ≤ Vol(β) ≤ 1} = χ{β: ≤Vol(β)≤1} dµn 2 2 1 ≥ φdµ = Σn−1φ(D#f iα) n n i=0 1 ≥ Σn−1χ (i). n i=0 α { ≤ ≤ } This shows by (2.1) that µn β : 2 Vol(β) 1 > 0forn large enough. Since { ≤ ≤ } # → → ∞ β : 2 Vol(β) 1 is a closed subset in Lk and µn µ0 as n + , we have µ {β : ≤ Vol(β) ≤ 1}≥lim → ∞µ {β : ≤ Vol(β) ≤ 1} > 0. 0 2 n + n 2 U # ≥ { ∈ # ≤ ≤ } So µ0( k (F )) µ0 β Lk : 2 Vol(α) 1 > 0. This together with (2.2) implies (2.3). Now we assert that there is a measure µ∗ ∈Awith ∗ # U ∩ ∗ µ ( k (F ) Eµ ) > 0 ∗ for some f−invariant and µ −full measure set Eµ∗ . Otherwise, we have for any µ ∈Aand any f−invariant and µ−full measure set Eµ, V# ∩ µ( k (F ) Eµ)=1,

and thus, by taking Eµ as Qµ, the basin of µ, it holds that V# ∩ µ( k (F ) Qµ)=1.

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V# ∩ # − − Set Fµ = k (F ) Qµ;thenitisD f invariant and of µ full measure. Then ∩ ⊂V# Qµ Fµ = Fµ k (F ), i.e., ∩ ∩U# ∅ Qµ Fµ k (F )= . This contradicts (2.3) and thus proves the assertion. Further, the ergodicity of µ∗ implies that # # ∗ # ∩ ∗ ⊂U Qµ (Lk ,D f) Eµ k (F ). # ∗ # ⊂ Note that since πQµ (Lk ,D f) Qm(M,f), we obtain # # ∗ # ∩ ∗ ⊂U Qµ (Lk ,D f) Eµ k (Qm(M,f)). ∗ ∈M U # ∗ So µ erg( k (Qm(M,f))). Without any confusion, we write µ as µ. Step III. The maximum property of the style number. M U # # Assume that there exists an ergodic measure µ in erg( ,D f)coveringm. U # → R For any frame α in Qµ, since the function τ : is continuous by Proposition 2.6, we have that n−1 1 # i lim → ∞ τ(D f α)= τdµ. n + n i=0 Qµ By Lemma 2.1, ≥ det(A(α)) 1 τdµ −1 dµ = −1 det(A(α))dµ > 0. Qµ Qµ Qµ So n−1 1 # i lim → ∞ τ(D f α) > 0. n + n i=0 According to Proposition 2.3 and Definition 1.3, we can deduce that ≤ k. Remark. In [2], Dai obtained similar results as the first part of Theorem 2.7 for C1 vector fields. ∗ Proof of Theorem 1.5. Let k = k+(m). We recall that F = Closure(Qm(M,f)). # # − Then Lk (F ) is compact and D f invariant. Take and fix ∈M U # µ erg( k (Qm(M,f))) # # ⊂ as in Theorem 2.7 to cover m. Then πQµ(Lk ,D f) Qm(M,f)and # # m(πQµ(L (F ),D f)) = 1. Now we prove Theorem 1.5(1)(2). The proof is an adaption of the proof to the main theorem about style number in [6]. { ∈ # | 1 } For every positive integer j we set Uj := γ Lk (F ) Vol(γ) > j and Wj := { ∈ # | ≥ 1 } γ Lk (F ) Vol(γ) j . Then

U1 ⊂ W1 ⊂···⊂Uj ⊂ Wj ⊂ Uj+1 ⊂··· . # → R ≤ ≤ Take by Urysohn’s Lemma a continuous function φj : Lk with 0 φj 1so # \ that φj(Wj)=1andφj(Lk Uj+1)=0. Then ≤ ≤ χWj φj χWj+1 ,

where χWj and χWj+1 denote the character functions for Wj and Wj+1, respectively.

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∈ # # ∈ U # # For each x πQµ(Lk ,D f)takeα Qµ( k ,D f)withπ(α)=x. Then we have

1 n−1 # i lim Σi=0 χα 1 (D f ) n→+∞ n j+1 1 n−1 # i = lim Σ χW (D f (α)) n→+∞ n i=0 j+1 ≥ 1 n−1 # i lim Σi=0 φj(D f α) n→+∞ n

= φjdµ ≥ χWj dµ

= µ(Wj) → 1,j→ +∞. ∗ ∗ − ∈ This implies that k+(x)=k+(m)=k for m a.e. x M. A similar argument shows 1 n−1 # i ≥ → → ∞ lim Σi=0 χα 1 (D f ) µ(Wj) 1,j + . n→−∞ |n| j+1 ∗ ≥ ∗ By Deﬁnition 1.2, it holds that k−(m) k+(m). ∗ ∗ ∗ Similarly for the case k = k−(m), we can obtain that k−(x)=k−(m)form − ∈ ∗ ≥ ∗ − ∈ ∗ a.e. x M and k+(m) k−(m). Consequently, for m a.e. x M, k−(x)= ∗ ∗ ∗ ∗ ∗ ∗ − ∈ k−(m),k+(x)=k+(m)andk−(m)=k+(m)=k (m). Moreover, for m a.e. x ∈U# M, there is α k (x) satisfying (1.1), i.e.,

1 n−1 # i 1 n−1 # i lim lim Σ χα(D f ) = lim lim Σ χα(D f )=1. →0 n→+∞ n i=0 →0 n→−∞ |n| i=0 ∈ # # This proves (1)(2). This also means that every α Qµ(Lk ,D f) satisﬁes (1.1) and # # − U # Qµ(Lk ,D f)isaµ full measure set in k . Hence we proved Theorem 1.5(3). 2.5. ProofofTheorem1.7.

Proof of Theorem 1.7. ∗ ∗ Step I. Find an ergodic measure m ∈Merg(M,f)sothatk (m)=k (M). ∗ # # Let k := k (M). We consider the topological system (Lk (M),D f). Denote by M # # # − inv(Lk ,D f)thesetofallD f invariant measures. By Deﬁnition 1.3, there − ∈U# { }+∞ are a k frame α k ,areal>0 and a subsequence nj j=0 of positive integers such that

1 nj −1 1 n−1 lim Σi=0 χα(i)=limn→+∞ Σi=0 χα(i) > 0. j→+∞ nj n Then it follows that 1 − lim Σnj 1Vol(D#f iα) → ∞ i=0 j + nj 1 − ≥ nj 1 # i # i (2.4) lim Σi=0 Vol(D f α)χα(D f ) j→+∞ nj 1 − ≥ nj 1 lim Σi=0 χα(i) > 0. j→+∞ nj

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Construct a sequence of measures µn satisfying j 1 − nj 1 # i ∀ ∈ 0 # R φdµnj := Σi=0 φ(D f α), φ C (Lk , ). nj { } By taking a subsequence when necessary, we can assume that µnj converge to some measure ∈M # # µ˜ inv(Lk ,D f). It follows by (2.4) that Voldµ˜ L# k

= lim Voldµnj j→+∞ # Lk 1 − = lim Σnj 1Vol(D#f iα) → ∞ i=0 j + nj > 0. ∈M # # From the Ergodic Decomposition Theorem (see [7]), there is µ erg(Lk ,D f) such that Voldµ>0. L#(M) k # U # ∪V# | U # Observe L = and VolV# =0. So U # Voldµ>0 and thus µ( ) > 0. k k k k k k U # ∈M U # # Since k is open and invariant, µ erg( k (M),D f). Let m := π∗(µ). We can check as in Theorem 2.7 and Theorem 1.5 that k∗(m)=k = k∗(M). ∗ U # Step II. Let k := k (M). We will find an ergodic measure µ0 on k with the largest U # independence number among all ergodic measures on k . M # # → R Defineτ ˜ : inv(Lk ,D f) byτ ˜(µ):= τdµ. Then the continuity of # → ∞ τ : Lk [0, + ) proved in Proposition 2.6 ensures thatτ ˜ is also a continuous ∗ # # function under the weak topology. Whenever µ ∈Merg(L ,D f), we have k 1 (2.5) τdµ = lim Σn−1τ(D#f iα)=˜τ(α), n→±∞ |n| i=1 ∀ ∈ # # M → R α Qµ(Lk ,D f), and thus τdµ=supα∈Qµ τ˜(α)=˜τ(µ). Soτ ˜ : inv is a natural extension ofτ ˜ : Merg → R in Definition 1.6. ∈ # { }∞ Let α Lk (M). Take nj j=1 such that by Definition 1.6 1 − τ˜(α) = lim Σnj 1τ(D#f iα). →∞ i=1 j nj

By taking a subsequence we assume that the measures µn deﬁned by j 1 − nj 1 # i ∈ 0 # R φdµnj = Σi=1 φ(D f α),φ C (Lk , ), nj

# # converge to some µ ∈Minv(L ,D f). Then k 1 nj −1 # i τ˜(µ)= τdµ = lim τdµnj = lim Σi=1 τ(D f α)=˜τ(α). j→+∞ j→∞ nj

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This implies by (2.5) that sup τ˜(µ)= sup τ˜(α). µ∈Minv ∈ # α Lk M # # → R Sinceτ ˜ : inv(Lk ,D f) is continuous under the weak* topology (see Propo- M # # ∈M # # sition 2.6) and inv(Lk ,D f) is compact, we can take µ1 inv(Lk ,D f)so that τ˜(µ1)= sup τ˜(µ)= sup τ˜(α). µ∈Minv ∈ # α Lk ∈M # # Claim I. There is µ0 erg(Lk ,D f)sothat

τ˜(µ0)=˜τ(µ1). In fact, otherwise, it could have # # τ˜(µ) < τ˜(µ1), ∀µ ∈Merg(L ,D f). k Set Q := Q (L#,D#f). Then µ(Q) = 1 for all µ ∈M (L#,D#f). µ∈Minv µ k inv k By the Ergodic Decomposition Theorem (see Theorem 2.16 in [7]), this deduces a contradiction as follows:

τ˜(µ1)= τdµ1 L# k

= τdµ dµ1 Q L# k

= τ˜(µ)dµ1 Q

< τ˜(µ1)dµ1 =˜τ(µ1). Q So Claim I follows. ∈M # # Take and ﬁx µ0 erg(Lk ,D f) such that

τ˜(µ0)= sup τ˜(α)= sup τ˜(µ). ∈ # µ∈Merg α Lk

Claim II. τ˜(µ0) > 0. ∗ Since k (m)=k for some m ∈Merg(M,f) by Step I, we can take µ2 ∈ M U # − ∈ erg( k )withπ∗(µ2)=m so that (1.1) in Theorem 1.5 follows for µ2 a.e. α U # k . So 1 n−1 # i # lim Σ χα(D f ) > 0,µ2 − a.e. α ∈U , n→∞ n i=1 k for small >0. By Proposition 2.3

1 n−1 # i # lim Σ τ(D f α) > 0,µ2 − a.e. α ∈U . n→∞ n i=1 k Since τ is continuous by Proposition 2.6 and µ2 is ergodic, we have by the Birkhoﬀ Ergodic Theorem,τ ˜(µ2) > 0. This impliesτ ˜(µ0) > 0. Claim II follows. ∈ # # Take and ﬁx α Qµ0 (Lk ,D f). Then by Claim II

1 n−1 # i 0 < τ˜(µ0) = lim Σ τ(D f α), n→∞ n i=1

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which implies by Proposition 2.3 that ∗ ∗ k (π∗µ0)=k (M).

Let m0 := π∗(µ0). Then m0 ∈Merg(M,f). The ergodic measures m0 and µ0 with π∗(µ0)=m0 meet the requirement in Theorem 1.7. This completes the proof.

3. Approximation property of invariant measures 3.1. Periodic shadowing. We start this subsection by introducing some prelimi- nary concepts and lemmas. Let f : M → M be a C1 diﬀeomorphism on a compact smooth Riemannian manifold M. + Pesin set ([11], [12]). Given λ, µ ε>0, and for all k ∈ Z , we deﬁne Λk = s u Λk(λ, µ; ε) to be all points x ∈ M for which there is a splitting TxM = E (x)⊕E (x) m s s m m u u m with invariant property Dxf (E (x)) = E (f x)andDxf (E (x)) = E (f x) satisfying: n εk −(λ−ε)n ε|m| (a) Df |Es(f mx)≤e e e , ∀m ∈ Z,n≥ 1; −n εk −(µ−ε)n ε|m| (b) Df |Eu(f mx)≤e e e , ∀m ∈ Z,n≥ 1; (c) tan(∠(Es(f mx),Eu(f mx))) ≥ e−εke−ε|m|, ∀m ∈ Z. +∞ We set Λ = Λ(λ, µ; ε)= k=1 Λk and call Λ a Pesin set. It is obvious that if ε1 <ε2,thenΛ(λ, µ; ε1) ⊆ Λ(λ, µ; ε2). Suppose f preserves an ergodic hyperbolic measureω. ˜ Thenω ˜ has s (s ≤ d = dimM) non-zero Lyapunov exponents

λ1 < ···<λr < 0 <λr+1 < ···<λs with an associated Oseledec splitting

TxM = E1(x) ⊕···⊕Es(x),x∈ O(˜ω), where O(˜ω) is the Oseledec basin ofω. ˜ We denote by

λ = |λr|,µ= λr+1, the absolute value of the largest negative Lyapunov exponent and the smallest positive Lyapunov exponent, respectively. Let s u E (x)=E1(x) ⊕···⊕Er(x),E(x)=Er+1(x) ⊕···⊕Es(x). Then we obtain a Pesin set Λ = Λ(λ, µ; ε)forasmallε.WecallitthePesin set associated withω. ˜ It follows (see, for example, Proposition 4.2 in [13]) that ω˜(Λ \ O(˜ω)) +ω ˜(O(˜ω) \ Λ) = 0. The following statements are elementary:

(a) Λ1 ⊆ Λ2 ⊆ Λ3 ⊆···; −1 (b) f(Λk) ⊆ Λk+1,f (Λk) ⊆ Λk+1; (c) Λk is compact for ∀k ≥ 1; u s (d) ∀k ≥ 1 the splitting x → E (x) ⊕ E (x) depends continuously on x ∈ Λk. { }+∞ Shadowing Lemma and Closing Lemma. Let δk k=1 be a sequence of positive { }+∞ real numbers. Let xn n=−∞ be a sequence of points in Λ=Λ(λ, µ, ε) for which { }+∞ there exists a sequence sn n=−∞ of positive integers satisfying: ∈ ∀ ∈ Z (a) xn Λsn , n ; (b) | sn − sn−1 |≤ 1, ∀n ∈ Z; ≤ ∀ ∈ Z (c) d(fxn,xn+1) δsn , n ;

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{ }+∞ { }+∞ ∈ then we call xn n=−∞ a δk k=1 pseudo-orbit. Given η>0,apointx M is an { }+∞ n ≤ ∀ ∈ Z η-shadowing point for the δk k=1 pseudo-orbit if d(f x, xn+1) ηεsn , n , −εk where εk = ε0e and ε0 are constants. Lemma 3.1 (Shadowing Lemma [4], [13]). Let f : M → M be a C1+α diffeomorphism, with a non-empty Pesin set Λ=Λ(λ, µ; ε) and ﬁxed parameters: { }+∞ λ, µ ε>0. For all η>0 there exists a sequence δk k=1 such that for any { }+∞ δk k=1 pseudo-orbit there exists a unique η-shadowing point. Lemma 3.2 (Closing Lemma [4]). Let f : M → M be a C1+α diﬀeomorphism and let Λ=Λ(λ, µ; ε) be a non-empty Pesin set. For all k ≥ 1, 0 <η<1,thereexists p p β = β(k, η) > 0 such that: if x, f x ∈ Λk and d(x, f x) <β, then there exists a p hyperbolic periodic point z ∈ M, with z = f z and d(z,x) <ηεk+1 <η.

We point out in the next lemma that the shadowing point in the Closing Lemma aboveisinthePesinsetΛ=Λ(λ, µ; ε)forsomeε >ε. A similar description of this result has appeared in [5], and thus we omit the proof. Lemma 3.3. Let f : M → M be a C1+α diffeomorphism preserving a hyperbolic ergodic measure ω˜ and let Λ=Λ(λ, µ; ε) be a non-empty Pesin set associated with ω˜. Then there is a positive number γ0 <εso that for any number 0 <γ<γ0,one can find a small number 0 <ν= ν(γ) ≤ ε with the following property: For any p k ≥ 1, 0 <η<1,thereisβ = β(k, η, ν) > 0 such that, if x, f x ∈ Λk(λ, µ; ν) and d(x, f px) <β, then there exists a hyperbolic periodic point z ∈ Λ(λ, µ; ν + γ) ⊂ p Λ(λ, µ; ε),withz = f z and d(z, x) <ηεk+1 <η. The following theorem and its corollary are about how to shadow finitely many orbit segments whose lengths are bounded or whose endpoints are limited into a given Pesin block, respectively, by a periodic orbit. Theorem 3.4. Let f : M → M be a C1+α diffeomorphism of a compact manifold and ω˜ be an ergodic hyperbolic measure with an associated Pesin set Λ=Λ(λ, µ; ε). ε Then there exists a positive number ν<2 with the following property: For any real η>0, any integer q ≥ 1, any sequence of points x1, ..., x ∈ Λ(λ, µ; ν) and any two sequences of integers a1 ≤ b1, ..., a ≤ b with 0 ≤ bi − ai ≤ q, there exist two ≥ ≥ − ∈ integers X 1 and p Σi=1(bi ai)+X, a hyperbolic periodic point z Λ and p a sequence of integers c1, ..., c such that f z = z and

cj +i i d(f z, f xj) <η, aj ≤ i ≤ bj,j=1, ..., . Proof. Take δ>0sothat η (3.1) d(x, y) <δ implies d(f ix, f iy) < 2 for any x, y ∈ M, any i =1, ..., q. For ε,takeγ0 > 0andfor0<γ<γ0,take ν = ν(γ) > 0 by Lemma 3.3. For any sequence of points

x1, ..., x ∈ Λ(λ, µ; ν), { }∞ { }∞ take by Lemma 3.1 a sequence δk k=1 so that each δk k=1 pseudo-orbit in the − η − { bi ai } Pesin set Λ(λ, µ; ν)canbe 2 traced. Let S = i=1 xi,f(xi), ..., f (xi) and

k0 = min{k ≥ 1 | S ⊂ Λk(λ, µ; ν)}.

Now we will consider Λk0 (λ, µ; ν). For simplicity, denote Λk0 =Λk0 (λ, µ; ν).

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{ } Take and ﬁx for Λk0 a ﬁnite open cover α = U1, ..., Ur by open balls Ui in M ∩ with diamUi <δk0 andω ˜(Ui Λk0 ) > 0,i=1, ..., r. Sinceω ˜ is ergodic, there exist positive integers Xij such that − Xij ∩ ∩ ∩ (3.2)ω ˜[f (Ui Λk0 ) (Uj Λk0 )] > 0.

Denote without confusion by Xij the least positive integer satisfying (3.2). ∈ ai ∈ bi ∈ Choose two open balls Ui0 ,Ui1 α with the property that f xi Ui0 ,f xi ∈ ∩ X(i+1)0i1 ∈ ∩ Ui1 ,i=1, 2, ..., . By (3.2) take yi Ui1 Λk0 so that f yi U(i+1)0 Λk0 − ∈ ∩ X101 ∈ ∩ for i =1, ..., 1. Take y U1 Λk0 so that f y U10 Λk0 . Hence we { }∞ obtain a δk k=1 periodic pseudo-orbit in the Pesin set Λ(λ, µ; ν) by repeating the following ﬁnite sequence inﬁnitely many times:

a1 b1 X2 1 −1 a2 b2 f x1, ..., f x1,fy1, ...,f 0 1 y1,f x2, ..., f x2,fy2, ..., (3.3) X3 2 −1 a b X1 −1 f 0 1 y2, ..., f x, ...., f x,fy, ..., f 0 1 y. Let

X =min{Xij}. 1≤i= j≤r The length of the ﬁnite sequence (3.3) is clearly larger than or equal to − Σi=1(bi ai)+X. By Lemma 3.1 and Lemma 3.3 there exists a unique peri- ∈ η − odic point z Λ, 2 -tracing the pseudo-orbit, with period p>Σi=1(bi ai)+X. This implies the existence of c1, ..., c which satisfy η d(f cj +iz, fix ) < ,a≤ i ≤ b ,j=1, ..., . j 2 j j This together with (3.1) completes the proof.

Corollary 3.5. Let f : M → M be a C1+α diﬀeomorphism of a compact manifold. Let ω˜ be an ergodic hyperbolic measure with an associated Pesin set Λ=Λ(λ, µ; ε).

ε Then there exists a positive number ν<2 with the following property: Given a real η>0 and an integer ≥ 1 with ω˜(Λ(λ, µ; ν)) > 0, there is an integer X = X(η) ≥ 1 such that for any ﬁnitely many points

x1, ..., xk ∈ Λ(λ, µ; ν)

ni and any positive integers n1, ..., nk,withf (xi) ∈ Λ(λ, µ; ν),i=1,...,k, there ∈ ≥ k exist a hyperbolic periodic point z Λ(λ, µ; ε), an integer p Σj=1nj + kX and a p cj +i i sequence of integers c1, ..., ck, such that f (z)=z, and d(f (z),f xj) <η, 1 ≤ i ≤ nj, 1 ≤ j ≤ k.

Proof. For ε>0, take γ0 > 0 and then take ν = ν(γ0) > 0 by Lemma 3.3. For { }+∞ { }+∞ the given real η>0, take a sequence δk k=1 by Lemma 3.1 so that each δk k=1 η − pseudo-orbit in the Pesin set Λ(λ, µ; ν)canbe 2 traced by a real orbit. For

given in the assumption, ﬁx δ. We replace δk0 by δ and Λ(λ, µ; ε)byΛ(λ, µ; ν) in the middle part of the proof of Theorem 3.4 and then a similar argument works; hence the conclusion holds.

For simplicity, we will identify the smaller Pesin set Λ(λ, µ; ν)withtheone Λ(λ, µ; ε) in the following proof.

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3.2. Density of atomic measures. Sigmund [14] proved that for uniformly hyperbolic diffeomorphisms with specification property, each invariant measure can be approximated by atomic measures. Hirayama [3] proved that each invariant measure supported by the closure of a Pesin set of a topologically mixing measure is approximated by atomic measures. Our result, Theorem 1.8 to be proved in this subsection, is to improve [3] by weakening the assumption of mixing measure to that of ergodic measure by applying Theorem 3.4 and Corollary 3.5 and the quanti- tative Poincaré Recurrence Lemma (i.e., Lemma 3.6). Moreover, the last inequality in Hirayama’s proof in [3] is not clear to us, since there is no estimation of − k nj 1 1 1 i 1 i (3.4) | sj( ξ(f (xj))) − ξ(f (z))| s nj #I j=1 i=0 i∈I (see (3.1) and (3.3) in [3] for the notations), which is necessary to complete the proof. We will give an explicit estimation in Theorem 1.8. Lemma 3.6. Let f : X → X be a homeomorphism of a compact metric space preserving an ergodic measure ω.˜ Let Γ ⊂ X be a measurable set with ω˜(Γ) > 0. Take γ>0. Then there exists a measurable function N(Γ, ·):X → N such that for ω˜ −a.e. x ∈ X, every n ≥ N(Γ,x) and every t ∈ [0, 1],thereissome ∈{0, 1, ..., n} ∈ | − | such that f (x) Γ and n t <γ. Proof. This is Lemma 3.12 in Bochi [1].

By using Lemma 3.6, we establish a lemma which will be applied to estimate (3.4). Lemma 3.7. Let f : X → X be a homeomorphism of a compact metric space preserving an ergodic measure ω.˜ Let Γj ⊂ X be measurable sets with ω˜(Γj) > 0, and for x ∈ Γj let r S(x, Γj):={r ∈ N | f x ∈ Γj},

j =1, ..., k. Take 1 >γ>0,T ≥ 1. Then for ω˜−a.e. xj ∈ Γj there exists nj = nj(xj) ∈ S(xj, Γj) such that nj ≥ T and

|n1 − nj| + ···+ |nj−1 − nj| + |nj+1 − nj| + ···+ |nk − nj| 0 < k <γ, Σj=1nj where j =1, ..., k. Proof. We prove the case when k =2andj = 1 and leave the others to the readers. 1 1 Take k0 > 1with ≤ γ ≤ . Take measurable functions N(Γi, ·):Γi → N k0 k0−1 as in Lemma 3.1, that is, forω ˜-a.e. xi ∈ Γi and every mi ≥ N(Γi,xi) and every i t ∈ [0, 1] there is some i ∈{0, 1, ..., mi}∩S(xi, Γi) such that | − t| <γ,i=1, 2. mi Take and fix a point xi ∈ Γi, which returns to Γi under positive iterations infinitely many times. Let us take and fix m2 ∈ S(x2, Γ2)sothat 1 1 − γ m > 2max{ (T +1),N(Γ ,x ),N(Γ ,x ), }. 2 1 − γ 1 1 2 2 γ Denote by [a] the largest integer not exceeding a. It holds that

k0 − 1 1 1 − γ [ m2] > max{ (T +1),N(Γ1,x1),N(Γ2,x2), }. k0 1 − γ γ

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k0−1 Using Lemma 3.1 by taking t =1andm1 =[ m2], there exists 1 ∈ k0 k0−1 {0, 1, ..., [ m2]}∩S(x1, Γ1)sothat k0 −γ< 1 − 1 <γ. k0−1 [ m2] k0 So k0 − 1 k0 − 1 [ m2](1 − γ) <1 < [ m2](1 + γ), k0 k0 and hence 1 >T.Then we have

k0−1 m − m − [ m2] 0 < 2 1 < 2 1 < 1 − k0 (1 − γ) 1 + m2 m2 m2 k0−1 m2 − 1 1 − γ < 1 − k0 (1 − γ) <γ+ < 2γ. m2 m2

Replacing 2γ by γ and denoting 1 by n1 and m2 by n2, we obtain that n2 − n1 n1,n2 >T and 0 < <γ. n1 + n2 This completes the proof. ∞ Proof of Theorem 1.8. Let Λ = =1 Λ be the Pesin set associated withω ˜.We | ˜ | denote byω ˜ Λ the conditional measure ofω ˜ on Λ. Set Λ = supp(˜ω Λ )and ˜ ∞ ˜ ±1 ˜ ⊂ ˜ s u Λ= =1 Λ. Clearly, f Λ Λ+1, and the sub-bundles E (x),E(x) depend continuously on x ∈ Λ˜ . Moreover, Λis˜ f−invariant withω ˜-full measure. We will show that Λ˜ meets the conditions of our theorem. Take and fix a real >0, an f−invariant measure ω ∈Minv(f)withsupp(ω) ⊂ Λ˜ and a finite set F of continuous functions, i.e., F ⊂ C0(M,R). We assume without | |≤ ∀ ∈ loss of generality that ξ 1, ξ F. Take 0 <η< 4 so that (3.5) d(x, y) <ηimplies |ξ(x) − ξ(y)| < , ∀x, y ∈ M, ∀ξ ∈ F. 8 Let 1 Q(f)={x ∈ M | the limit exists lim Σn−1ξ(f ix)=ξ∗(x), ∀ξ ∈ C0(M,R)}. n→+∞ n i=0 By the Birkhoff Ergodic Theorem, ω(Q(f)) = 1 andω ˜(Q(f)) = 1. For x ∈ Q(f) there exists N = N(x) such that 1 (3.6) | Σn−1ξ(f ix) − ξ∗(x)| < , n i=0 8 whenever n ≥ N(x). We divide the following proof into several steps. Step I. Constructing a partition of Q(f). Let A =sup{|ξ∗(x)|; x ∈ Q(f),ξ∈ F }. Recall that by [8A] we refer to the ∈ { ∈ maximal integer not exceeding 8A. For j =1, ..., [8A]+1 ,ξ F, set Qj(ξ)= x − (j−1) ≤ ∗ − j} B { } Q(f), A + 8 ξ (x) < A + 8 . Let := ξ∈F Q1(ξ), ..., Q[8A]+1(ξ) . Then B is a partition of Q(f). For η chosen above, define {δ}≥1,δ <ηby Lemma 3.1. Take an integer 0 ∩ ˜ − ∩ ˜ − large enough so thatω ˜(Q(f) Λ0 ) > 1 16 and ω(Q(f) Λ0 ) > 1 16 . Also, ˜ { } take and fix for Λ0 a finite cover α = U1, ..., Ur by open balls in M so that

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∩ ˜ diam(Ui) <δ0 andω ˜(Ui Λ0 ) > 0,i=1, ..., r. We can assume without loss of generality that r ω( Uj )=1. j=1 B Whenever an element B in is contained not inside a single set Ui in α but γ instead in a union of open sets, say, j=1 Uj, we then divide B into γ subsets as follows:

γ−1 B1 = B ∩ U1,B2 = B ∩ (U2 \ U1), ..., Bγ = B ∩ (Uγ \ Uj ). j=1

{ }k In this way we deﬁne a partition for Q(f), denoted by Qj j=1. Observe that for ∈ ⊂ each Qj,thereisUj α such that Qj Uj and thus diam(Qj) <δ0 ,j=1, ..., k. Without loss of generality we may assume that ω(Qj) > 0foreachj =1, ..., k and k thus may assume that ω( j=1 Qj)=1. ∈ Step II. A reduction of Q(f) ξdω, ξ F. ⊂ ∩ ˜ By using Poincar´e’s Recurrence Lemma, for Qj take a subset Nj Qj Λ0 such that

(1) ω(Nj)=˜ω(Nj)=0; ∈ ∩ ˜ \ nj ∈ ∩ ˜ (2) for xj (Qj Λ0 ) Nj there exists an integer nj so that f xj Qj Λ0 ; (3)

1 − nj 1 i ∗ | | Σi=0 ξ(f xj)=ξ (xj)+τj, τj < ,j=1, ..., k. nj 8 ∗ k ∗ From Step I, | ξ dω − Σ ξ (xj) dω| < ;thuswehave Q(f) j=1 Qj (f) 8 1 − | ξdω− Σk ω(Q ) Σnj 1ξ(f ix )| j=1 j n i=0 j Q(f) j (3.7) | − k ∗ | = ξdω Σj=1ω(Qj)(ξ (xj)+τj) Q(f) < + = ,ξ∈ F. 8 8 4 ∈ Step III. A further reduction of Q(f) ξdω, ξ F. ≥ 1 η For η>0 as in (3.5) take S>0 such that s S implies s < k . Fix an integer s ≥ S and takes ¯1, ..., s¯k

s¯ s¯ +1 j ≤ ω(Q ) ≤ j . s j s

It follows by taking sj =¯sj ors ¯j +1that

s 1 η s =Σk s , |ω(Q ) − j | < < . j=1 j j s s k

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Thus from (3.7) and the choice of η it follows that 1 1 − | ξdω− Σk s Σnj 1ξ(f ix )| s j=1 j n i=0 j Q(f) j 1 − 1 (3.8) ≤| − k nj 1 i | ξdω Σj=1ω(Qj) Σi=0 ξ(f xj) + k Q(f) nj s = + η<2 = ,ξ∈ F. 4 4 2 Step IV. Constructing a pseudo-orbit and a tracing periodic point. Recall thatω ˜ is ergodic, and thus for each pair (i, j), 1 ≤ i, j ≤ k,thereis ≥ X(i,j) ∩ ∩ ˜ ∅ ∈ ∩ ˜ an integer X(i, j) 1 such that f Qi Qj Λ0 = . Take yi Qi Λ0 so X(i,i+1) ∈ ∩ ˜ ≤ X(k,1) ∈ ∩ ˜ that f yi Qi+1 Λ0 ,1 i

n1−1 n1−1 n1−1 X(1,2)−1 x1, ..., f x1,x1, ..., f x1, ..., x1, ..., f x1,y1, ..., f y1,

s1 times

n2−1 n2−1 n2−1 X(2,3)−1 x2, ..., f x2,x2, ..., f x2, ..., x2, ..., f x2,y2, ..., f y2,

(3.9) s2 times ......

nk−1 nk−1 nk−1 X(k,1)−1 xk, ..., f xk,xk, ..., f xk, ..., xk, ..., f xk,yk, ..., f yk.

sk times By Corollary 3.5 there exists a hyperbolic periodic point z ∈ Λ with period p, η-tracing this pseudo-orbit. Then k ··· − p =Σj=1sjnj + X(1, 2) + X(2, 3) + + X(k 1,k)+X(k, 1). Let 1 p−1 µ = Σ δ i , z p i=0 f z where δ denotes the Dirac measure supported at the point x.Then x 1 − ξdµ = Σp 1ξ(f iz), ∀ ξ ∈ F. z p i=0 Step V. Completing the proof. n −1 n −1 We denote by [xj,f j xj] the orbit segment xj, ..., f j xj in (3.9) and let

k sj nj I = [xj,f xj]. j=1 i=1 Denote by Γ the set

{0, 1, ..., n1 − 1,n1, ..., 2n1 − 1, ..., (s1 − 1)n1, ..., s1n1 − 1,

s1n1 + X(1, 2), ..., s1n1 + n2 + X(1, 2) − 1, ...,

s1n1 +(s2 − 1)n2 + X(1, 2), ..., s1n1 + s2n2 + X(1, 2) − 1, ...... , k−1 k−1 k−1 k−1 − Σj=1 sjnj +Σj=1 X(j, j +1), ..., Σj=1 sjnj +Σj=1 X(j, j +1)+nk 1, ..., k−1 k−1 − k k−1 − } Σj=1 sjnj +Σj=1 X(j, j +1)+(sk 1)nk, ..., Σj=1sjnj +Σj=1 X(j, j +1) 1 .

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k Then #I =Σj=1sjnj =#Γ. By (3.2), we have

1 1 i | Σ ∈ ξ(x) − Σ ∈ ξ(f z)|≤ . #I x I #Γ i Γ 8 Observe that |ξ|≤1, ∀ξ ∈ F ; hence | p−1 i − i |≤ Σi=0 ξ(f z) Σi∈Γξ(f z) k max X(i, j). 1≤i,j≤r Thus we have

1 p−1 i 1 | Σ ξ(f z) − Σ ∈ ξ(x)| p i=0 #I x I

1 p−1 i 1 i ≤| Σ ξ(f z) − Σ ∈ ξ(f z)| + p i=0 #Γ i Γ 8 (3.10) 1 p−1 i i 1 i = | Σ [#Γ ξ(f z) − pξ(f z)] + Σ ∈ ξ(f z)| + p #Γ i=0 #Γ i/Γ 8 p(p − #Γ) k max ≤ ≤ X(i, j) 2k max ≤ ≤ X(i, j) ≤ + 1 i,j r + ≤ 1 i,j r + . p #Γ #Γ 8 #Γ 8 Hence

1 p−1 i 1 (3.11) | Σ ξ(f z) − Σ ∈ ξ(x)| < ,ξ∈ F, p i=0 #I x I 4

by choosing nj large enough. Without loss of generality, we assume nj are taken for xj by Lemma 3.7, j = 1, 2, ..., k. It follows that

1 − 1 1 − | p 1 i − k nj 1 i | Σi=0 ξ(f z) Σj=1sk Σi=0 ξ(f xj) p s nj

1 p−1 i 1 (3.13) = | Σ ξ(f z) − Σ ∈ ξ(x)| + p i=0 #I x I 4 < ,ξ∈ F. 2

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Combining (3.8) with (3.13) we obtain that 1 1 − 1 − | ξdω− ξdµ |≤| Σk s Σnj 1ξ(f ix ) − Σp 1ξ(f iz)| + z s j=1 j n i=0 j p i=0 2 (3.14) Q(f) j ≤ + = , ∀ξ ∈ F. 2 2 This completes the proof.

4. Approximation properties on mean angle, independence number and Oseledec splittings In this section we prove Theorem 1.12. We ﬁrst introduce a theorem due to Sun and Vargas [17] which is one of the keys in the proof of Theorem 1.12. U # − U # → Recall that is a bundle of unit frames, π : M is the canonical M U # # # − bundle projection and erg( ,D f)isthesetofallD f invariant and ergodic U # ≤ ≤ measures on , 1 d = dimM. Theorem 4.1 ([17]). Let f : M → M be a C1 diﬀeomorphism preserving an ergodic measure ν with simple spectrum. For a given integer , 1 ≤ ≤ d = dimM, B { ∈M U # # | } denote := µ erg( ,D f) π∗(µ)=ν .Then ≤ B ≤ d Ad Card( ) 2 Ad, − ··· − where Ad = d(d 1) (d +1). # U # ∪V# Recall that Ld = d d . Now we discuss the relation between the Oseledec splitting of a measure ν ∈Merg(M,f) and a generic frame of a measure µ ∈ M # # erg(Ld ,D f)coveringν, i.e., π∗(µ)=ν. Suppose ν has a simple spectrum and its Oseledec splitting is

TM = E1 ⊕ E2 ⊕···⊕Ed,d= dimM. We claim that for any point x ∈ O(ν), the Oseledec basin of ν,andanyframe ∈ # # ∈ α =(u1(x), ..., ud(x)) Qµ(Ld ,D f), it holds that ui(x) Ej(i)(x)forsome ∀ # → R j(i), i =1, ..., d. Indeed, we deﬁne φi : Ld by ∈ # φi(β)=log πiDf(β) ,βLd ,

where πi denotes the projection from a frame to its i−th vector, i =1, 2, ..., d. Then all these functions φi are clearly bounded and continuous. By the deﬁnition # # of Qµ(Ld ,D f) and the Birkhoﬀ Ergodic Theorem, we have n 1 n 1 # j lim log Df (ui) = lim φi(D f α)= φidµ, n→±∞ n n→±∞ n j=1

which implies that ui ∈ Ej(i) for some j(i), and thus the claim follows. In light of Theorem 4.1 and this claim, it seems reasonable to classify all the U # ergodic measures on d which cover an ergodic measure with a simple spectrum. V# Since the measures supported on d make no contribution to mean angles or independence numbers (for details, see Step II in the proof of Theorem 1.12(1)), we will U # take into account the measures on d only. In order to make a precise classiﬁcation in the proof of Theorem 1.12, we need a new notion as follows.

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Deﬁnition 4.2. Let ν be an ergodic measure on M with simple spectrum:

λ1 <λ2 < ···<λd,d= dimM.

∈M U # # An ergodic measure µ erg( d ,D f)coveringν is said to be (σ(1),σ(2),..., σ(d))−type if

log πiDfdµ = λσ(i),i=1, 2, ..., d, U # d where σ : {1, 2, ..., d}→{1, 2, ..., d} is a permutation.

Remark. If µ is (σ(1),σ(2), ..., σ(d))−type, by the claim before Deﬁnition 4.2, one U # # can deduce that any frame α =(u1,u2, ..., ud) in the basin Qµ( d ,D f)must satisfy that ui ∈ Eσ(i) for any i =1, 2, ..., d,where

TM = E1 ⊕ E2 ⊕···⊕Ed

is the Oseledec splitting of ν.Wealsosayα to be of (σ(1),σ(2), ..., σ(d))−type in this case.

1 ≥ Proof of Theorem 1.12. Let ε = n ,n 1. Applying Theorem 1.9 and Theorem 1.8 to the ergodic measureω ˜, we can ﬁnd a hyperbolic periodic point zn with period pn which satisﬁes the following properties:

(a) the Lyapunov exponents of zn are ε−close to those of the measureω ˜ in the sense that after arranging all the Lyapunov exponents of zn and those ofω ˜ in increasing order, respectively, the i−th exponent of zn and the i−th one ofω ˜ are ε−close, i =1, 2, ..., d; 1 pn−1 (b) the atomic measure ωn = δ i is ε−close to the measureω ˜ in pn i=1 f zn the weak*-topology.

Without loss of generality we assume that ωn has a simple spectrum asω ˜ has for all n ≥ 1.

Proof of Theorem 1.12(1). We need to verify that there exists a convergence sub- { } sequence znk k≥1 such that

p −1 nk 1 m m lim ∠(Ei(f (zn )),Et(f (zn ))) = m∠ω˜ (Ei,Et), 1 ≤ i = t ≤ d, k→∞ p k k nk m=0

m ⊕···⊕ m m where E1(f (znk )) Ed(f (znk )) is the Oseledec splitting at f (znk ). We divide the proof into three steps.

# Step I. Finding an appropriate invariant measure µ on Ld to cover the measureω ˜. ∈M U # # Choose an ergodic measure µn erg( d ,D f)coveringωn according to U # ⊆ # # Theorem 2.7. Since d Ld and Ld is compact, we can assume that µn converges ∈M # # to an invariant measure µ inv(Ld ,D f) by taking subsequence if necessary. Then µ is a covering measure ofω ˜.Thatis,π∗µ =˜ω, since given any continuous

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function ξ : M → R, it follows that ξdω˜ = limn→+∞ ξdωn → ∞ ξdπ∗µ = limn + n → ∞ ξ ◦ πdµ = limn + n ξ ◦ πdµ = = ξdπ∗µ. Therefore, we get the following commutative graph: ∈M U # # / ∈M # # µn erg( d ,D f) µ inv(Ld ,D f)

π∗ π∗ / ωn ∈Merg(M, f) ω˜ ∈Merg(M, f). By the Birkhoﬀ Ergodic Theorem, j−1 1 # i (4.1) φdµ = lim φdµn = lim lim φ(D f αz ), n→+∞ n→+∞ j→+∞ j n i=0 ∀ ∈ U # # ∀ ∈ b U # R αzn Qµn ( d ,D f), φ C ( d , ). Step II. Analyzing the number and type of the ergodic measures decomposed from µ. Let A { ∈M U # # | } := ν erg( d ,D f) π∗(ν)=˜ω . Claim. The ergodic measures ν decomposed from µ are all supported on the bundle U # ∈A d , i.e., ν . Proof. Denote H(·):L#(M) →F#(M) to be the Gram-Schmidt orthogonalization # # ∪···∪ # F # process and normalization where L (M)=L1 (M) Ld (M)and (M)= F # ∪···∪F# F # # 1 (M) d (M). Each bundle i (M), a subset of Li (M), consists of all the orthogonal frames with i vectors, 1 ≤ i ≤ d. Denote ω =(ω0,ω1, ..., ωn, ...)whereω0 =˜ω.For1≤ k ≤ d =dimM,let F # k (ω) { ∈F# || | ∃ ∈U# ∃ ≥ := α =(u1,...,uk) k (M) ui =1, β =(v1,...,vk) k (M), n 0,

s.t. β is a quasi-regular point of some covering measure of ωn and α = H(β)}.

Note that each ωn has a simple spectrum and thus its style number is d, and note ≤ U # k d. Theorem 2.7 ensures the existence of ergodic measures on k (M)covering ≥ F # ωn, n 0. Hence the bundle k (ω) is well deﬁned. Let # { ∈ # || | ∃ ≥ Lk (ω):= α =(u1, ...,uk) Lk (M) ui =1, n 0,s.t.

ui is contained in some Oseledec invariant bundle of ωn, ∀ i =1, 2, ..., k}. # The vectors of any frame in Lk (ω) are not necessarily taken from diﬀerent Oseledec bundles. Denote # # ∪···∪ # F # F # ∪···∪F# L (ω)=L1 (ω) Ld (ω)and (ω)= 1 (ω) d (ω).

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∞ They are two bundles supported on i=0 Qωi (M, f)whereQωi (M, f) is the basin of ωi, i ≥ 0. Also, let # # # χ = H ◦ Df|F #(ω) : F (ω) →F (ω). Then we get a commutative graph:

D#f / L#(ω) L#(ω)

H H χ# / F #(ω) F #(ω).

It is clear that H : L#(ω) →F#(ω) is a continuous surjection and the measures # µn and µ are all supported on the bundle L (ω). Let

µñ := H∗µn andµ ˜ := H∗µ. # Then they are all supported on F (ω)and˜µn → µ˜ as n →∞.Moreover,sinceµn U # F # are supported on d , it must hold thatµ ñ are supported on d (ω). Note thatµ ñ covers ωn andµ ˜ coversω ˜, respectively, and ωn → ω˜ as n → +∞. We can deduce F #| thatµ ˜ is supported on d Qω˜ (M, f). According to the Ergodic Decomposition Theorem and the result in [17] (the same result as Theorem 4.1 for F #(M)), there exist finitely many ergodic measures m˜ , ..., m˜ ∈M (F #| ,χ#)andpositiverealnumbersa , ..., a with 1 erg d Qω˜ (M, f) 1 ≤ ≤ d i=1 ai =1(1 2 d!) such that

µ˜ = a1m˜ 1 + ···+ am˜ . −1 F #| ⊆U# Since H ( d Qω˜ (M, f)) d (M), all the measures coveringµ ˜ must be supported U # ∈M U # on d (M). Especially, we have that µ erg( d (M)).Henceweprovedthe claim.

Combining the claim with Theorem 4.1, the number of ergodic measures decom- d U # posed from µ is not more than 2 d! and they are all supported on d . Since adding ergodic measures weighing vacancy to the sum of all ergodic measures decomposed from µ will not have any effect, we assume that the set of ergodic measures decomposed from µ coincides with A and CardA =2dd!.Nowwearrangeallthese d d 2 d! measures, denoted by νj, j =1, 2, ..., 2 d!, according to their types (see Defi- nition 4.2). We first divide all these 2dd!measuresintod sets, each having 2d(d−1)! members and the −th set consisting of all the measures of (, ∗, ∗, ···∗)−type, =1, 2, ..., d. Specifically, the −th set is

d d {νj ∈A|νj is of (, ∗, ∗, ···∗) − type ,j=( − 1) · 2 (d − 1)! + 1, ..., · 2 (d − 1)!}, where ( − 1) · 2d(d − 1)! and · 2d(d − 1)! denote the sum of the cardinalities of the ﬁrst − 1 sets and the sum of the cardinalities of the ﬁrst sets, respectively. Further, we divide each set into d−1 subsets with cardinalities all being 2d(d−2)! such that the i−th subset of the −th set contains all the measures of (, i, ∗, ∗, ··· ∗ − V# , ) type, i =1, 2, ..., d,andi = (a measure should be supported on d =

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# −U# ∗ ∗ ···∗ − − − Ld d if it is (, , , , ) type). More precisely, the i th subset of the th set is

{νj ∈A|νj is of (, i, ∗, ∗, ···∗) − type , j =( − 1) · 2d(d − 1)! + (i − 1)2d(d − 2)! + 1, ..., ( − 1) · 2d(d − 1)! + i · 2d(d − 2)!}. Here the number (i−1)2d(d−2)! denotes the sum of the cardinalities of the first i−1 subsets in the −th set, and hence the number ( − 1) · 2d(d − 1)! + (i − 1)2d(d − 2)! denotes the sum of the cardinalities of the first −1 sets and the cardinalities of these i−1 subsets. Analogically, one explains the number (−1)·2d(d−1)!+i·2d(d−2)!. By repeating this procedure, one can divide A into d! disjoint classes, each containing 2d measures. The question as to which class a measure in A belongs d to is uniquely determined by its type. Specially, the first 2 measures, νj,j= 1, 2, ..., 2d, are of (1, 2, 3,...,d)−type (here ‘···’ represents the sequence of integers from 4 to d − 1 in increasing order). Using the Ergodic Decomposition Theorem, there exists a function ι : A→[0, 1] such that ν∈A ι(ν)=1and 2dd! ∀ ∈ b U # R (4.2) φdµ = ( φdνj)ι(νj), φ C ( d , ), U # # j=1 Qνj ( d ,D f) U # # where Qνj ( d ,D f) denotes the basin of νj (see Section 1 for the definition). Let d aj = ι(νj), 1 ≤ j ≤ 2 d!.

Applying the Birkhoﬀ Ergodic Theorem to νj in (4.2), it holds for any αj ∈ U # # Qνj ( d ,D f)that 2dd! n−1 1 # i (4.3) φdµ = aj( lim φ(D f (αj))), n→+∞ n j=1 i=0 ∈ b U # R for all φ C ( d , ). Combining (4.1) with (4.2), we have j−1 2dd! n−1 1 # i 1 # i (4.4) lim lim φ(D f αz )= aj( lim φ(D f (αj))), n→+∞ j→+∞ j n n→+∞ n i=0 j=1 i=0 ∈ b U # R ∈ U # # ∈ U # # for all φ C ( d , ),αzn Qµn ( d ,D f), αj Qνj ( d ,D f). We need the following proposition, saying that there are at most 2d ergodic measures summing up in the right hand of (4.2). Proposition 4.3. Under the conditions of Theorem 1.12, the number of the ergodic measures decomposed from µ is not more than 2d and these ergodic measures are of the same type.

Proof. Take φt for φ in (4.4), where φt,t=1, 2, ..., d,are deﬁned before Deﬁnition 4.2. Then j−1 2dd! n−1 1 # i 1 # i (4.5) lim lim φt(D f αz )= aj( lim φt(D f (αj))). n→+∞ j→+∞ j n n→+∞ n i=0 j=1 i=0

Denote by {λ1(zn) <λ2(zn) < ··· <λd(zn)} and {λ1 <λ2 < ··· <λd} the Lyapunov spectrum of ωn and that ofω ˜, respectively. We have lim λt(zn)=λt, n→+∞

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t =1, ..., d, by Theorem 1.9. An analogous discussion as in the above claim before 1 j−1 # i Deﬁnition 4.2 shows that the limit lim φt(D f αz )coincideswithsome j→+∞ j i=0 n Lyapunov exponent of ωn, assumed to be λt(zn), the t−th one, t =1, 2, ..., d. Combining with (4.5), we obtain

2dd! n−1 1 # i (4.6) λt = aj( lim φt(D f (αj))). n→+∞ n j=1 i=0 U # Since ω ˜ has a simple spectrum and νj is supported on d , it holds that φt1 dνj and φ dν are diﬀerent for any 1 ≤ t = t ≤ d. A similar argument for ν as t2 j 1 2 j 1 n−1 # i in the above claim shows that limn→+∞ n i=0 φt(D f (αj)) coincides with the − ∈ U # # Lyapunov exponent of the t th vector in the frame αj Qνj ( d ,D f). In the following we will give a detailed computation on (4.6) when t varies from 1 to d and then complete the proposition inductively. When t =1,wehave

2d(d−1)! 2·2d(d−1)! 3·2d(d−1)! 2dd! λ1 = λ1 aj + λ2 aj + λ3 aj + ···+ λd aj , j=1 j=2d(d−1)!+1 j=2·2d(d−1)!+1 j=(d−1)·2d(d−1)!+1 i.e.,

2d(d−1)! 2·2d(d−1)! 3·2d(d−1)! 2dd! λ1(1 − aj )=λ2 aj + λ3 aj + ···+ λd aj . j=1 j=2d(d−1)!+1 j=2·2d(d−1)!+1 j=(d−1)·2d(d−1)!+1 2dd! Since j=1 aj =1, it follows that

2·2d(d−1)! 3·2d(d−1)! 2dd! λ1 aj + λ1 aj + ···+ λ1 aj j=2d(d−1)!+1 j=2·2d(d−1)!+1 j=(d−1)·2d(d−1)!+1

2·2d(d−1)! 3·2d(d−1)! 2dd! = λ2 aj + λ3 aj + ···+ λd aj . j=2d(d−1)!+1 j=2·2d(d−1)!+1 j=(d−1)·2d(d−1)!+1 Thus we have (4.7) 2·2d(d−1)! 3·2d(d−1)! 2dd! (λ1 −λ2) aj +(λ1 −λ3) aj +···+(λ1 −λd) aj =0. j=2d(d−1)!+1 j=2·2d(d−1)!+1 j=(d−1)·2d(d−1)!+1 d Since λ1 <λ2 < ···<λd, and 0 ≤ aj ≤ 1, 1 ≤ j ≤ 2 d!, (4.7) implies that

2·2d(d−1)! 3·2d(d−1)! 2dd! aj = aj = ···= aj =0. j=2d(d−1)!+1 j=2·2d(d−1)!+1 j=(d−1)·2d(d−1)!+1 So d aj =0, ∀ j ≥ 2 (d − 1)! + 1 and thus 2d(d−1)! (4.8) aj =1. j=1

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When t = 2, we have by (4.8) that

2d(d−2)! 2·2d(d−2)! 3·2d(d−2)! 2d(d−1)! λ2 = λ2 aj + λ3 aj + λ4 aj + ···+ λd aj . j=1 j=2d(d−2)!+1 j=2·2d(d−2)!+1 j=(d−2)·2d(d−2)! From (4.8), a similar argument as in (4.7) shows that

2·2d(d−2)! 3·2d(d−2)! 2d(d−1)! (λ2 −λ3) aj +(λ2 −λ4) aj +···+(λ2 −λd) aj =0. j=2d(d−2)!+1 j=2·2d(d−2)!+1 j=(d−2)·2d(d−2)!

Since λ2 < ···<λd,wehave 2·2d(d−2)! 3·2d(d−2)! 2d(d−1)! aj = aj = ···= aj =0. j=2d(d−2)!+1 j=2·2d(d−2)!+1 j=(d−2)·2d(d−2)! Thus the following equations are true: 2d(d−2)! d (4.9) aj =1andaj =0,j≥ 2 (d − 2)! + 1. j=1 Proceeding analogously to consider the case for t =3, 4, ..., d, we can ﬁnally prove that 2d d (4.10) aj =1andaj =0,j≥ 2 +1. j=1 Hence, the proposition is proved. Proposition 4.3 tells us that the sum in (4.2) is actually taken for 2d ergodic measures all having the same type. Hence we ﬁnish the proof of Step II, and in the next step we focus on these 2d ergodic measures.

Remark.In the proof of Proposition 4.3 we assumed that the limit 1 j−1 # i lim φt(D f αz )coincideswithλt(zn), the t−th Lyapunov exponent of j→+∞ j i=0 n ωn,whereφt and λt(zn) share the same index t, t =1, 2, .., d. The other cases without this assumption can be proved similarly. For instance, assume d = dimM =3 and j−1 1 # i lim lim φt(D f αz )=λ2t mod (3) (here let λ0 = λ3), n→+∞ j→+∞ j n i=0

t =1, 2, 3, where we recall λ1 <λ2 <λ3. We first discuss the case when t =2 (i.e., the above limit is λ1), then t = 1 (the above limit is λ2) and finally the case when t = 3 (the above limit is λ3), one after another. Then we can also prove that therearenotmorethan23 ergodic measures decomposed from µ and they are all (2, 1, 3)−type. Step III. Completing the proof of (1). U # → R → ∠ Define φ0 : 2 , (u1,u2) (u1,u2). Clearly, φ0 is a bounded con- U # ∈{ } U # →U# tinuous function on 2 .Fori, j 1, 2, ..., d , we define πij : d 2 , → U # → R (u1,u2, ..., ud) (ui,uj), and φij : d ,φij(α)=φ0(πij(α)).

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Let φ = φit,1≤ i = t ≤ d in (4.4); then by (4.10)

p −1 d − 1 n 2 1 n1 (4.11) lim φ (D#f sα )= a ( lim φ (D#f s(α ))). → ∞ it zn j → ∞ it j n + pn n + n s=0 j=1 s=0

d Since all the 2 measures νj are of the same type by Proposition 4.3 and thus all the d 2 frames αj in (4.11) are of the same type by the Remark following Deﬁnition 4.2, # s s s we have φit(D f (αj)) = ∠(Df Ei,Df Et), which does not depend on the choice of j.Thus − pn 1 1 s s lim ∠(Df Ei(zn),Df Et(zn)) n→+∞ p n s=0 − pn 1 1 # s = lim φit(D f αz ) n→+∞ p n n s=0 2d n−1 1 s s = aj( lim ∠(Df Ei,Df Et)) n→+∞ n j=1 s=0 n−1 1 s s = lim ∠(Df Ei,Df Et) n→+∞ n s=0

= m∠ω˜ (Ei,Et).

So the mean angles between the Oseledec invariant bundles Ei(zn)andEt(zn) converge to the mean angle between the Oseledec invariant bundles Ei and Et.We have proved Theorem 1.12(1).

Proof of Theorem 1.12(2). We recall the measures ωn andω ˜ in (a)(b) and µn and µ in the proof of (1) and their relations:

ωn → ω,˜ µn → µ, π∗µn = ωn,π∗µ =˜ω. U # → R → Since the function τ : d is continuous by Proposition 2.6, τdµn τdµ. Since all the ergodic measures decomposed from µ have the same type, we have that τdµ= τdω.˜

Note that ωn is an ergodic measure with simple spectrum. This implies that

τdµn = τdωn as shown in the explanation below (1.2). Hence we have that τdωn → τdω˜ or τ˜(ωn) → τ˜(˜ω). We thus obtain (2).

Proof of Theorem 1.12(3). Take k large, so that

(4.12)ω ˜(Λk(˜ω)) > 1 − ε,

where Λk(˜ω) denotes the k−th Pesin block associated withω ˜. Since the splitting

x → E1(x) ⊕···⊕Ed(x)

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depends continuously on x ∈ Λk(˜ω), we can choose a uniform constant η>0 satisfying

For each x ∈ Λk(˜ω) ∩ Supp(˜ω), we take and ﬁx α0(x)=(u1, ..., ud) ∈ U # # ∩ Qµ( d (x),D f) Supp(µ), where µ is the measure taken in the proof of (1). Denote by U(α0(x),η)theη−neighborhood of α0(x) under the metric on the Grass- man bundle. Then µ(U(α0(x),η)) > 0. Recalling that µn → µ,wehave

(4.14) lim inf µn(U(α0(x),η)) ≥ µ(U(α0(x),η)) > 0. n→+∞

Therefore, we can take an integer N(x)=N(α0(x)) > 0 such that

(4.15) µn(U(α0(x),η)) > 0, ∀ n ≥ N(x). U # # ∩ ∩ This implies the existence of a frame βn(x)inQµn ( d ,D f) supp(µn) U(α0(x),η)foreachn ≥ N(x). Observing that µn covers the atomic measure ωn, we can deduce that βn(x) must be a frame based on a periodic point on Orb(zn), where zn is the periodic point chosen in (a)(b). Since ωn is a hyperbolic measure with simple spectrum by Theorem 1.9, we have by the claim before Definition 4.2 that each vector in βn(x) must be in a subbundle of the Oseledec splitting of ωn and any two vectors lie in different subbundles. Thus the Oseledec splitting ofω ˜ at x is η closed by the Oseledec splitting of ωn at a point z = z(x, n)on Orb(zn),n≥ N(x). Note that the number N(x)mayvarywithx; we need to find ˜ ∈ anumberN, independent of the choice of x Λk(˜ω), such that µN˜ meets (3). This can be done by the compactness of Λk(˜ω) and continuity of the Oseledec splitting on it. Hence we complete the proof of Theorem 1.12.

Proof of Theorem 1.13. (a) is from Katok’s closing lemma [4]. (b) is from Theorem 1.9. (c) is from Theorem 1.8. Combining (a)-(c) with Theorem 1.12, one can deduce (d)-(f) immediately.

Acknowledgement The authors thank the referee for his/her suggestions for improving the manu- script.

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LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, Peo- ple’s Republic of China – and – Applied Mathematical Department, The Central Uni- versity of Finance and Economics, Beijing 100081, People’s Republic of China E-mail address: [email protected] E-mail address: [email protected] LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, Peo- ple’s Republic of China – and – China Institute of Advanced Study, The Central Uni- versity of Finance and Economics, Beijing 100081, People’s Republic of China E-mail address: [email protected] LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, Peo- ple’s Republic of China E-mail address: [email protected]

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