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

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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 hyper- bolic 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 er- godic 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1544 CHAO LIANG, GENG LIU, AND WENXIANG SUN # # 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use APPROXIMATION PROPERTIES ON INVARIANT MEASURE 1545 Let X be a metric space. We denote by Cb(X, R) the set of all bounded con- tinuous 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1546 CHAO LIANG, GENG LIU, AND WENXIANG SUN 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.
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