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<ul><li><p>Journal of Dynamical and Control Systems, Vol. 12, No. 4, October 2006, 465488 ( c2006)</p><p>ERGODIC HYPERBOLIC ATTRACTORSOF ENDOMORPHISMS</p><p>DA-QUAN JIANG and MIN QIAN</p><p>Abstract. Let be an SRB-measure on an Axiom A attractor of a C2-endomorphism (M, f). As is known, -almost every x is positively regular and the Lyapunov exponents of (f, Tf) at x are</p><p>constants (i)(f, ), 1 i s. In this paper, we prove that Lebesgue-almost every x in a small neighborhood of is positively regular andthe Lyapunov exponents of (f, Tf) at x are the constants (i)(f, ),1 i s. This result is then generalized to nonuniformly com-pletely hyperbolic attractors of endomorphisms. The generic propertyof SRB-measures is also proved.</p><p>1. Introduction</p><p>Assume that is a hyperbolic attractor of a C2 Axiom A dieomorphism(M,f) and that v is the Lebesgue measure on M induced by the Riemannianmetric. It is well known that v-almost all x in the basin of attraction W s()are generic with respect to the SRB-measure + of f on (see [3, Theorem4.12]), i.e.,</p><p>limn+</p><p>1n</p><p>n1</p><p>i=0</p><p>fix = +. (1)</p><p>If is an ergodic invariant measure of a C2-dieomorphism g, the Lyapunovexponents of g not zero are -almost everywhere and if the measure hasthe SRB-property (i.e., the conditional measures of on unstable manifoldsare absolutely continuous with respect to the corresponding Lebesgue mea-sures), then Ledrappier [15] and Pugh and Shub [24] proved that the setof points generic with respect to has positive Lebesgue measure. Thisgeneric property of SRB-measures is of a particular interest for physics</p><p>2000 Mathematics Subject Classification. 37D20, 37D25, 37C40.Key words and phrases. Hyperbolic attractor, endomorphism, Lyapunov exponent,</p><p>SRB-measure, absolute continuity of local stable manifolds.This work was supported by the 973 Funds of China for Nonlinear Science, the NSFC</p><p>10271008, and the Doctoral Program Foundation of the Ministry of Education.</p><p>465</p><p>1079-2724/06/1000-0465/0 c 2006 Springer Science+Business Media, Inc.</p><p> DOI:10.1007/s1088300600021</p></li><li><p>466 DA-QUAN JIANG and MIN QIAN</p><p>(see [8, 9, 33]). It makes easy to compute the space averages of various ob-servables approximately via their time averages, even if the SRB-measureis singular, since the initial point can be chosen in the basin of attractionof the attractor uniformly with respect to the Lebesgue measure.</p><p>For the hyperbolic attractor of the Axiom A dieomorphism (M,f),by the Oseledec multiplicative ergodic theorem, +-almost every x isLyapunov regular and the Lyapunov exponents of (f, Tf) at x are constants(i)(f, +), 1 i s. That is, there exists a linear decomposition of TxM ,TxM = U</p><p>(1)x U (s)x satisfying the condition</p><p>limn</p><p>1nlog Txfnu = (i)(f, +)</p><p>for all 0 = u U (i)x , 1 i s. Jiang et al. [12] proved that v-almost everyx W s() is positively regular and the Lyapunov exponents of (f, Tf) at xare the constants (i)(f, +), 1 i s. That is, there exists a sequence oflinear subspaces of TxM , {0} = V (0)x V (1)x V (s)x = TxM satisfyingthe condition</p><p>limn+</p><p>1nlog Txfnu = (i)(f, +)</p><p>for all u V (i)x \V (i1)x , 1 i s. Jiang et al. [12] also showed that a similarresult holds for a general nonuniformly completely hyperbolic attractor withan ergodic SRB-measure. Tsujii [36] asserted that an ergodic probabilitymeasure of a dieomorphism f without zero Lyapunov exponents is anSRB-measure if and only if the set of points, which are generic with respectto and positively regular with the same constant Lyapunov exponents asthose associated with , has a positive Lebesgue measure. However, theproof of the suciency is somewhat not easy to access, and the detailedproof of the necessity was given by Jiang et al. [12].</p><p>In practical applications, one should choose an initial point when com-puting approximately the Lyapunov exponents. The above large ergodicproperty of the Lyapunov exponents associated with an SRB-measure jus-ties that the initial point can be taken in the basin of attraction of theattractor uniformly with respect to the Lebesgue measure while what onekeeps in mind is the Lyapunov exponents with respect to the SRB-measure.In general, the hyperbolic attractor may have a fractal structure and a sin-gular SRB-measure and, therefore, the Lebesgue measure is a more prefer-able reference measure for sampling the initial point than the SRB-measure,although the Lebesgue measure is, in general, not an invariant measure.</p><p>For a random hyperbolic dynamical system generated by small pertur-bations of a deterministic Axiom A dieomorphism, Liu and Qian [19],and Liu [18] proved that its SRB-measure has a similar generic property asabove; while Jiang et al. [13] showed that the Lyapunov exponents of therandom system have a similar large ergodic property as above.</p></li><li><p>ERGODIC HYPERBOLIC ATTRACTORS OF ENDOMORPHISMS 467</p><p>The purpose of this paper is to use the methods developed in the abovereferences to attack the case of endomorphisms.</p><p>Let M be a smooth, compact, and connected Riemannian manifold with-out boundary and v be the Lebesgue measure on M induced by the Rie-mannian metric. Let O be an open subset of M and O be an Axiom Aattractor of an endomorphism f C2(O,M) (see Sec. 2 below for the def-inition of an Axiom A attractor). Qian and Zhang [25] showed that thereexists a unique f -invariant Borel probability measure on satisfying thePesin entropy formula:</p><p>h(f) =</p><p>s(x)</p><p>i=1</p><p>(i)x +m(i)x d(x),</p><p>where h(f) is the measure-theoretic entropy of f with respect to , and (1)x < (2)x < < (s(x))x < + are the Lyapunov exponents of(f, Tf) at x with the multiplicities m(i)x , 1 i s(x). Qian and Zhang [25]also proved that if > 0 is suciently small and the set of critical pointsCf = {x O|det(Txf) = 0} has zero Lebesgue measure, then for v-almostall x B(),</p><p>limn+</p><p>1n</p><p>n1</p><p>i=0</p><p>fix = , (2)</p><p>where B() = {y O | d(y,) < }. Since the SRB-measure is f -ergodic, the Lyapunov spectrum of (f, Tf) is -almost everywhere equal toa constant</p><p>{((i)(f, ),m(i)(f, )) : 1 i s}.In Sec. 2, after we review some basic notions and results about Axiom Aendomorphisms, we exploit the absolute continuity of local stable manifoldsand the SRB property of to prove the following result.</p><p>Theorem 1. Let f C2(O,M) and O be an Axiom A attractor off , and suppose that Txf is nondegenerate for every x . Then there exists > 0 such that Lebesgue-almost every x B() is positively regular andthe Lyapunov spectrum of (f, Tf) at x is a constant {((i)(f, ),m(i)(f, )) :1 i s}. That is, there exists a sequence of linear subspaces of TxM ,{0} = V (0)x V (1)x V (s)x = TxM satisfying the condition</p><p>limn+</p><p>1nlog Txfnu = (i)(f, )</p><p>for all u V (i)x \ V (i1)x , 1 i s. In addition, dimV (i)x dimV (i1)x =m(i)(f, ), 1 i s.</p><p>The results of (2) and Theorem 1 can also be generalized to the caseof nonuniformly completely hyperbolic attractors of endomorphisms. More</p></li><li><p>468 DA-QUAN JIANG and MIN QIAN</p><p>concretely, suppose that f is a C2-endomorphism of a compact Riemannianmanifold M . Zhu [39] introduced the inverse limit space of (M,f) to over-come the diculty arising from the noninvertibility and improved the localunstable manifold theorem of Pugh and Shub [24] for (M,f). Qian, Xie,and Zhu [26,27] presented a formulation of the SRB-property for an invari-ant measure of the endomorphism (M,f) and proved that this propertyis sucient and necessary for the Pesin entropy formula:</p><p>h(f) =</p><p>M</p><p>s(x)</p><p>i=1</p><p>(i)x +m(i)x d(x).</p><p>Assume that is f -ergodic, then for -almost every x M , the Lyapunovspectrum {((i)x ,m(i)x ) : 1 i s(x)} of (f, Tf) at x is equal to a constant:</p><p>{((i)(f, ),m(i)(f, )) : 1 i s}.In Sec. 3, we employ the absolute continuity of local stable manifolds toprove the following result.</p><p>Theorem 2. Suppose that is an ergodic invariant measure of the C2-endomorphism (M,f) satisfying the following conditions:</p><p>1. log |det(Txf)| L1(M,);2. is an SRB-measure of (M,f);3. the Lyapunov exponents of (f, Tf) are -almost everywhere not zero,</p><p>moreover, the smallest Lyapunov exponent</p><p>(1)(f, ) = min{(i)(f, ) : 1 i s} < 0.Then there exists a Borel set M such that f = , () = 1 and thatfor every x W s() def= </p><p>yW s(y),</p><p>limn+</p><p>1n</p><p>n1</p><p>i=0</p><p>fix = ,</p><p>where W s(y) is the global stable set of f at y and, moreover, v(W s()) > 0.</p><p>Every point x W s() \+n=0</p><p>fnCf is positively regular and the Lyapunov</p><p>spectrum of (f, Tf) at x is the constant {((i)(f, ),m(i)(f, )) : 1 i s},where the set of critical points Cf = {y M |det(Tyf) = 0}. If, in addition,v(Cf ) = 0, then v(W s() \</p><p>+n=0</p><p>fnCf ) > 0.</p><p>Conversely, we wonder whether the existence of a positive Lebesgue mea-sure set of points with genericity and positive regularity as above impliesthe SRB-property of , as asserted by Tsujii [36] in the situation of dieo-morphisms.</p></li><li><p>ERGODIC HYPERBOLIC ATTRACTORS OF ENDOMORPHISMS 469</p><p>The above results justify that for uniformly or nonuniformly completelyhyperbolic attractors of endomorphisms, the initial points can be chosenclose to the attractors uniformly with respect to the Lebesgue measures,to compute the space averages of observables approximately via their timeaverages, or to compute approximately the Lyapunov exponents associatedwith the SRB-measures.</p><p>In applications, if the dierentiable mappings or the equations of motionthat dene dynamical systems are completely known, Lyapunov exponentsare computed by a straightforward technique using a phase space plus tan-gent space approach (see [2, 8, 10,34,37]).</p><p>2. Lyapunov exponents of Axiom Aattractors of endomorphisms</p><p>The purpose of this section is to prove Theorem 1. But rst, we needto review some basic notions and results about hyperbolic sets of endomor-phisms.</p><p>We begin with the notion of an inverse limit space. Let X be a compactmetric space and T be a continuous mapping on X. The inverse limit spaceXT of the system (X,T ) is dened as the subset of XZ consisting of all fullorbits, i.e.,</p><p>XT ={x = {xi}iZ</p><p> xi X, Txi = xi+1 i Z}.</p><p>Obviously, XT is a closed subset of XZ which is endowed with the producttopology and the metric</p><p>d(x, y) =</p><p>iZ</p><p>12|i|</p><p>d(xi, yi) x, y XZ.</p><p>Let p be the natural projection from XT to X, i.e., p(x) = x0 for all x XT ,and : XT XT be a left-shift homeomorphism, then p = T p. Denotethe set of all T -invariant Borel probability measures on X byMT (X), whilethe notation M (XT ) has a similar meaning. The projection p induces acontinuous bijection between M (XT ) and MT (X) as follows:</p><p>p() = ( p) M (XT ), C(X);(XT , , ) is ergodic if and only if (X,T, p) is ergodic.</p><p>Throughout this paper, we assume that M is a smooth, compact, andconnected Riemannian manifold without boundary and denote by v theLebesgue measure on M induced by the Riemannian metric. Assume thatf is a C1-endomorphism on M , i.e., f C1(M,M). Write E = pTM forthe pullback bundle of the tangent bundle TM by the projection p, and</p><p>Ex = pxTMppx</p><p>Tx0M</p></li><li><p>470 DA-QUAN JIANG and MIN QIAN</p><p>for the natural isomorphisms between bers Ex and Tx0M :</p><p> = (x, v)ppx</p><p>v (x Mf , v Tx0M).</p><p>A ber-preserving mapping on E with respect to can be dened as</p><p>px Tf p : Ex Exfor each x Mf . For convenience, we will still denote it by Tf . Tf isa linear mapping on each ber and there is a constant K > 0 such thatTf(x) K for all x Mf .</p><p>Let O be an open subset of M , f C1(O,M), and = f() O be acompact invariant set of f . Since is f -invariant, the inverse limit spacef of the system (, f), the natural projection p from f to , and theleft-shift homeomorphism on f can be dened as above. From later onin this section, we also denote</p><p>Mf = {x = {xi}iZ : xi O, f(xi) = xi+1 i Z}.A compact f -invariant set is said to be hyperbolic if there exists a con-tinuous splitting pTM = E|f = Es Eu such that</p><p>1. Tf(Es) Es, Tf(Eu) Eu;2. Tfn(s) An0s s Es, n Z+,Tfn(u) A1n0 u u Eu, n Z+,</p><p>where 0 (0, 1) and A 1 are constants. For each x f , the decompo-sition of the ber Ex = Esx Eux may depend on the past. For x = y withx0 = y0, it may happen that pEux = pEuy . However, pEsx depends onlyon x0.</p><p>A compact f -invariant set is called an Axiom A basic set of f if1. is hyperbolic;2. P (f), where P (f) is the set of periodic points of f ;3. f is topologically positively transitive on , i.e., there exists x </p><p>such that Orb+(x) = {f ix}+i=0 is dense in ;4. there exists an open set V satisfying </p><p>iZf i(V ) = .</p><p>An Axiom A basic set of f is called an Axiom A attractor if there existarbitrarily small open neighborhoods U of such that fU U .</p><p>Example 1. Let A be a (d d)-matrix with elements in Z and absolutedeterminant greater than 1, all of whose eigenvalues have absolute valuedierent from 1. Then the linear mapping A on Rd induces a multi-to-one endomorphism fA on the d-dimensional torus Td = Rd/Zd, as in thesituation of linear Anosov dieomorphisms on torus. For each x Td, deneEsx, E</p><p>ux as TfA-invariant subspaces of TxT</p><p>d associated with eigenvalues ofabsolute value less than 1 and greater than 1, respectively. Then one cansee that the whole torus Td is a hyperbolic set of fA. Applying the local</p></li><li><p>ERGODIC HYPERBOLIC ATTRACTORS OF ENDOMORPHISMS 471</p><p>product structure and shadowing property presented in [25, 32] to fA, onecan prove that the periodic points of fA are dense in Td, like the Anosovclosing lemma [3] in the dieomorphism situation. Hence Td is an Axiom Aattractor of fA. Obviously, fA preserves the Lebesgue measure on Td. Forexample,</p><p>A =(n 11 1</p><p>), n 3,</p><p>with eigenvalues</p><p>0 < 1 =n+ 1n2 2n+ 5</p><p>2< 1 < 2 =</p><p>n+ 1 +n2 2n+ 52</p><p>,</p><p>induces a hyperbolic endomorphism on T2.</p><p>Let be a hyperbolic invariant set of f Cr(O,M) (r 1) andE|f = Es Eu be the hyperbolic splitting. For each x f and > 0,we write</p><p>Esx() = {es Esx : es < },Eux () = {eu Eux : eu < },Ex() = Esx() Eux ().</p><p>For small > 0, x , and x f , the local stable and unstable manifoldsare dened, respectively, as follows:</p><p>W s (x)def={y M d(fnx, fny) < ,n Z+},</p><p>Wu (x)def={y0 M</p><p> y Mf such thaty0 = p(y), d(xn, yn) < n Z+</p><p>}.</p><p>For x f , the global unstable set Wu(x) of f at x in Mf is dened by</p><p>Wu(x) def={y Mf</p><p> lim supn+</p><p>1nlog d(yn, xn) < 0</p><p>},</p><p>and the global unstable set of f at x in M is dened by Wu(x) def= pWu(x).Let</p><p>Wu (x) ={y Mf d(yn, xn) < n Z+</p><p>},</p><p>which is called a local unstable set of f at x in Mf . Then p : Wu (x) Wu (x) is bijective. For x , the global stable set W s(x) of f at x in Mis dened by</p><p>W s(x) def={y M</p><p> lim supn+</p><p>1nlog d(fny, fnx) < 0</p><p>}.</p></li><li><p>472 DA-QUAN JIANG and MIN QIAN</p><p>One has</p><p>Wu(x) =+</p><p>n=0</p><p>fn(Wu (nx)) x f ,</p><p>W s(x) =+</p><p>n=0</p><p>fn(W s (fnx)) x .</p><p>(3)</p><p>If is an Axiom A attractor of f , then there exists a suciently small > 0such that Wu (x) x f [25, Proposition 2.1], and hence</p><p>Wu(x) = +n=0fn(Wu (nx)) x f .Denote by Bx() the open ball on M of radius centered at x M ,</p><p>and by Bx() the open ball on Mf of radius centered at x Mf . Bythe property of the continuous splitting E|f = Es Eu, there exists aconstant a > 0 such that for any e = eseu with es Es, eu Eu, one has</p><p>max{es, eu} a</p><p>2e.</p><p>The following proposition is a result of changing coordinates from the the-orem about stable and unstable manifolds for hyperbolic invariant sets ofendomorphisms (see [25, Theorems 2....</p></li></ul>

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