Ergodic Hyperbolic Attractors of Endomorphisms

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  • Journal of Dynamical and Control Systems, Vol. 12, No. 4, October 2006, 465488 ( c2006)

    ERGODIC HYPERBOLIC ATTRACTORSOF ENDOMORPHISMS

    DA-QUAN JIANG and MIN QIAN

    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

    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.

    1. Introduction

    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.,

    limn+

    1n

    n1

    i=0

    fix = +. (1)

    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

    2000 Mathematics Subject Classification. 37D20, 37D25, 37C40.Key words and phrases. Hyperbolic attractor, endomorphism, Lyapunov exponent,

    SRB-measure, absolute continuity of local stable manifolds.This work was supported by the 973 Funds of China for Nonlinear Science, the NSFC

    10271008, and the Doctoral Program Foundation of the Ministry of Education.

    465

    1079-2724/06/1000-0465/0 c 2006 Springer Science+Business Media, Inc.

    DOI:10.1007/s1088300600021

  • 466 DA-QUAN JIANG and MIN QIAN

    (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.

    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

    (1)x U (s)x satisfying the condition

    limn

    1nlog Txfnu = (i)(f, +)

    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

    limn+

    1nlog Txfnu = (i)(f, +)

    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].

    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.

    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.

  • ERGODIC HYPERBOLIC ATTRACTORS OF ENDOMORPHISMS 467

    The purpose of this paper is to use the methods developed in the abovereferences to attack the case of endomorphisms.

    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:

    h(f) =

    s(x)

    i=1

    (i)x +m(i)x d(x),

    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(),

    limn+

    1n

    n1

    i=0

    fix = , (2)

    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

    {((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.

    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

    limn+

    1nlog Txfnu = (i)(f, )

    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.

    The results of (2) and Theorem 1 can also be generalized to the caseof nonuniformly completely hyperbolic attractors of endomorphisms. More

  • 468 DA-QUAN JIANG and MIN QIAN

    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:

    h(f) =

    M

    s(x)

    i=1

    (i)x +m(i)x d(x).

    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:

    {((i)(f, ),m(i)(f, )) : 1 i s}.In Sec. 3, we employ the absolute continuity of local stable manifolds toprove the following result.

    Theorem 2. Suppose that is an ergodic invariant measure of the C2-endomorphism (M,f) satisfying the following conditions:

    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,

    moreover, the smallest Lyapunov exponent

    (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=

    yW s(y),

    limn+

    1n

    n1

    i=0

    fix = ,

    where W s(y) is the global stable set of f at y and, moreover, v(W s()) > 0.

    Every point x W s() \+n=0

    fnCf is positively regular and the Lyapunov

    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() \

    +n=0

    fnCf ) > 0.

    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.

  • ERGODIC HYPERBOLIC ATTRACTORS OF ENDOMORPHISMS 469

    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.

    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]).

    2. Lyapunov exponents of Axiom Aattractors of endomorphisms

    The purpo

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