活血定眩丸的毒理学研究

Commun. Math. Phys. 211, 687 – 703 (2000) Communications in Mathematical Physics © Springer-Verlag 2000 Computing Invariant Densities and Metric Entropy Mark Pollicott1, Oliver Jenkinson2 1 Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. E-mail: mp@ma.man.ac.uk 2 UPR 9016 CNRS, Institut de Mathématiques de Luminy, 163 avenue de Luminy, case 907, 13288 Marseille, cedex 9, France. E-mail: omj@iml.univ-mrs.fr Received: 25 July 1999 / Accepted: 7 January 2000 Abstract: We present a method for accurately computing the metric entropy (or, equiv- alently, the Lyapunov exponent) of the absolutely continuous invariant measure  for a piecewise analytic expanding Markov map T of the interval. We construct atomic signed measures M supported on periodic orbits up to period M , and prove thatR log jT 0j dM ! h./ super-exponentially fast. We illustrate our method with several examples. 0. The Problem Let T V I ! I be a piecewise C2 expanding map of the interval. By the Lasota- Yorke theorem [L-Y] we know there exists a T -invariant probability measure  which is absolutely continuous with respect to Lebesgue measure. If T is topologically mixing then this absolutely continuous invariant measure (abbreviated to a.c.i.m.) is unique and ergodic, with strictly positive C1 density function  V I ! R. In the special case of the continued fraction transformation T x D 1=x (mod 1) the density is .x/ D ..1 C x/ log 2/−1. For the Ulam-von Neumann map T x D 4x.1 − x/, which is topologically conjugate to the expanding tent map .x/ D 1 − j2x − 1j, the density is known to be .x/ D −1.x.1 − x//−1=2. Finally, Parry [Pa] and Renyi [Renyi] studied the problem for certain piecewise linear examples, including the - transformation defined by T x D x (mod 1), for any  > 1. However, in general there is no explicit formula for the density function, and much recent interest has focused on approximating the density [HuntB,Froy1,Froy2,HuntF,K-M-Y,M], with particular emphasis on Ulam’s method [Ulam]. In this note we will present an alternative approach. We shall only consider the case where T is a real analytic Markov map. In this case a well-known approach to finding the a.c.i.m.  is given by the weighted distribution of periodic orbits. More precisely, the sequence of atomic T -invariant probability measures mM D P x2Fix.M/ x=j.T M/0.x/jP x2Fix.M/ 1=j.T M/0.x/j ; M  1 .0:1/ 688 M. Pollicott, O. Jenkinson (where the summations are over the set Fix.M/ D fx 2 I V T Mx D xg of period-M points) converges to  in the weak-star topology (see [K-H, p. 635]). That is, for each k 2 Z, Z 1 0 e2ikxdmM.x/ ! O.k/ VD Z 1 0 e2ikxd.x/: This convergence can be shown to be exponentially fast (see Sect. 4), the rate being determined by the second eigenvalue of the Perron-Frobenius operator LT (defined in Sect. 1). In this note we will present a rather more efficient method of computing these Fourier coefficients O.k/. We achieve this by a more elaborate regrouping of the periodic points to define new invariant signed probability measures M by M D 1 NM X .n1;::: ;nm/ n1C:::CnmM .−1/m mW mX iD1 X x2Fix.ni / 0 B@ mY jD1 j 6Di X z2Fix.nj / r.z; nj / 1 CA xj.T ni /0.x/ − 1j ; where we denote r.x; n/ D 1 nj.T n/0.x/ − 1j ; and the normalisation constant NM is simply NM D X .n1;::: ;nm/ n1C:::CnmM .−1/m mW mX iD1 X x2Fix.ni / 0 B@ mY jD1 j 6Di X z2Fix.nj / r.z; nj / 1 CA 1j.T ni /0.x/ − 1j : Note in particular that the measures M are supported on those periodic points of period at most M . The advantage of these more involved definitions is that in the case when T is C! we have the following superexponential convergence of the Fourier coefficients. Theorem 1. Suppose that T V I ! I is a piecewise C! expanding Markov map of the interval, with absolutely continuous invariant probability measure . There is a sequence of T -invariant signed probability measures M , supported on the points of period at most M , and 0 <  < 1, such that for each k 2 Z, there exists C > 0 with Z 1 0 e2kixdM.x/ − O.k/   CM2 : More generally, for any real analytic function g V I ! R, we can similarly see that j R 10 g.x/dM.x/ − R 10 g.x/d.x/j  CM2 . Of particular interest is the choice g.x/ D log jT 0.x/j, since by the Rohlin–Pesin equality [Ro,Pe] we can identify the metric entropy h.T / with such an integral, h.T / D Z log jT 0.x/jd.x/: .0:2/ The ergodic theorem means that h.T / equals the Lyapunov exponent lim n!C1 1 n log j.T n/0.x/j for Lebesgue almost all x. Combining identity (0.2) with the method of proof of Theorem 1, we also prove Computing Invariant Densities and Metric Entropy 689 Theorem 2. Suppose that T V I ! I is a piecewise C! expanding Markov map of the interval, with absolutely continuous invariant probability measure . There is a sequence of T -invariant signed probability measures M , supported on the points of period at most M , and constants 0 <  < 1, C > 0, such that jh.T /− R log jT 0j dM j  CM2 . We organise the article as follows. In Sect. 1 we collect together a few easy re- sults on expanding piecewise-analytic Markov interval maps, and consider analytically parametrised families of such maps. In Sect. 2 we show that the measures M converge to the a.c.i.m. . In Sect. 3 we consider the speed of convergence, and prove our the- orems. In Sect. 4 we use our methods to estimate the entropy of the a.c.i.m. for two families of examples. 1. Some Simple Properties of the Density and Metric Entropy Consider a piecewise C! expanding Markov interval map T V I ! I . By this we mean (a) there exists a partition 0 D a0 < a1 < : : : < ap D 1 such that T is real-analytic on each Tai; aiC1U (i.e., extends to a holomorphic map on some complex neighbourhood of Tai; aiC1U), (b) there exists  > 1 such that jT 0.x/j   for all x 2 I . We shall also assume for convenience that T Tai; aiC1U D I for all i. This condition is not essential, but eases the exposition in the sequel. In particular it implies T is topologically mixing. A more general definition of a Markov map would allow each image T Tai; aiC1U to be some union [j2J Taj ; ajC1U of pieces in the partition. Under this hypothesis, the invariant density  in Lemma 1 would only be C! on each piece of the partition. Moreover, the various function spaces we consider would be defined relative to the partition, and would consist of functions with some prescribed regularity on each piece of the partition. Let  denote the unique ergodic a.c.i.m. [C-E] and let  2 L1.I / denote the associated density (or Radon-Nikodym derivative) i.e., d.x/ D .x/d.x/, where  is Lebesgue measure. In general it is not possible to find an explicit expression for the density . We can define the Perron–Frobenius operator LT V C0.I / ! C0.I / by LT k.x/ D X TyDx k.y/ jT 0.y/j ; k 2 C 0.I /: Recall that the space of real-analytic functions C!.I/ is a Fréchet space. If we fix some open complex neighbourhood U of I , then the space of bounded holomorphic functions on U is a Banach space with respect to the uniform norm. Lemma 1. The density  is a real-analytic function, and satisfies L D . This is a standard result following from the change of variable formula (cf. [K-H, p. 186]), the latter observation following immediately from the fact that L preserves C!.I/, i.e., LC!.I/  C!.I/. Lemma 2. Suppose the metric entropy h.T / is equal to the topological entropy h.T /. Then the map T is C! conjugate to a piecewise linear map whose slope on each partition piece is p, where p is the number of pieces in the Markov partition. 690 M. Pollicott, O. Jenkinson A simple calculation with Radon-Nikodym derivatives [K-H, p. 187] gives dT d .x/ D d.T / d.T / d.T / d d d D .T x/jT 0.x/j.x/−1: .1:1/ By T -invariance of  the jacobian dT=d must satisfy X TyDx  dT d .y/ −1 D 1; a.e. .1:2/ If h.T / D h.T /, then  is the unique measure of maximal entropy, so (see [Wal1]) − log jT 0j is an essential coboundary. Thus − log dT d is also an essential coboundary (i.e. − log dT d D vT − v C c for some real c and bounded measurable function v). But then (1.2) implies dT d must itself be constant, and that this constant is precisely p. Therefore, if we assume that 0 is a fixed point, for convenience, then by (1.1) we deduce T is conjugate to the required piecewise linear map by the conjugacy h.x/ DR x 0 .y/dy. Lemma 3. Let T" (−a < " < a) be a family of C! expanding Markov maps, which vary analytically with the parameter ", and with a.c.i.m. ". Then the associated metric entropies h."/ also vary analytically. The invariant density " for T" satisfies LT"" D " (by Lemma 1). Moreover, the maximal eigenvalue 1 for LT" is simple and isolated. It follows from standard analytic perturbation theory that the map " 7! " 2 C!.I/ is C! [Kato]. By (0.2) we can write h."/ D R log jT 0" j".x/dx, from which Lemma 3 immediately follows. Let us now consider the specific family of maps T" V T0; 1U ! T0; 1U defined by T".x/ D 2x C " sin 2x (mod 1); for − 12 < " < 1 2 : These maps are all analytic and expanding. They are also topologically conjugate to the standard doubling map T0 V I ! I , from which we deduce that the topological entropy h.T"/ is equal to log 2 (indeed the same is true for any degree-2 expanding map [K-H, Theorem 2.4.2]). By the variational principle (see for example [Wal2]), the metric entropy of the a.c.i.m. " must necessarily satisfy h."/  log 2. The following lemma is an easy consequence of Lemma 2. Lemma 4. For " 6D 0 the map T" has metric entropy h."/ < log 2. If we assume for a contradiction that h."/ D log 2 then Lemma 1 implies that T" is C! conjugate to the linear map with derivative 2. However, when we compute the derivative at the fixed point 0 we see T 0".0/ D 2.1 − "/, whereas it is required to take the value T 0".0/ D 2, giving the required contradiction. Indeed we can derive a second order approximation of h."/ as follows. Proposition 1. h."/ D log 2 − 24 "2 C O."3/ as " ! 0. Computing Invariant Densities and Metric Entropy 691 Proof. The Rohlin–Pesin identity gives h."/ D Z log jT 0".x/j".x/dx; .1:3/ where " is the invariant density for T". By Lemma 3, we know the entropy h."/ varies analytically, so let us write h."/ D log 2 Ch1" Ch2"2 CO."3/, where the coefficients h1, h2 are to be determined. We have the expansions log jT 0".x/j D log 2 C " cos.2x/ − "22 2 cos2.2x/ C O."3/ and ".x/ D 1 C 1.x/" C 2.x/"2 C O."3/: The functions 1, 2 satisfy R 1.x/ dx D 0 D R 2.x/ dx, since R ".x/ dx D 1 for all ". Substituting the expansions into (1.3), and comparing coefficients, we immediately deduce that h1 D 0. Comparing second order terms we have h2 D log 2 Z 2.x/dx C  Z cos.2x/1.x/dx −  2 2 Z cos2.2x/dx: .1:4/ The first term on the right-hand side of (1.4) is zero, by the previous comment. We now claim that the function 1 is identically zero. Once we have established this, the second term on the right-hand side of (1.4) will vanish to leave h2 D − 2 2 Z cos2.2x/dx D −2=4; as required. Let " be the inverse branch of T" which is close to x 7! x=2. We can write ".x/ D x=2 C " 1.x/ C O."2/, and using the identity x D T".".x//, we derive  1.x/ D − 12 sin.x/. If " is the inverse branch near x 7! .x C 1/=2, then we can write ".x/ D .xC1/=2C" 1.x/CO."2/, and a similar argument gives  1.x/ D 12 sin.x/ D − 1.x/: Since " is a fixed point of the Perron–Frobenius operator LT" we have ".x/ D LT"".x/ D "."x/ 0".x/ C "."x/ 0".x/: .1:5/ Substituting the various series expansions into (1.5), and considering the linear term in ", we obtain 1.x/ D 1 2  1 x 2  C 1  x C 1 2  C . 1/0.x/ C . 1/0.x/ D 1 2  1 x 2  C 1  x C 1 2  : We thus deduce that 1 D LT01, so that 1 lies in the eigenspace spanned by the invariant density for the doubling map T0. That is, 1 is a constant. Having already observed that R 1.x/dx D 0, we finally deduce that 1 is identically zero, as required. ut In Fig. 1 we compare h."/ to the second order approximation derived in Proposi- tion 1. 692 M. Pollicott, O. Jenkinson -0.1 -0.05 0.05 0.1 0.665 0.67 0.675 0.68 0.685 0.69 Fig. 1. Entropy h."/ and second order approximation log 2 − 24 2 (dashed) 2. Convergence of the Measures M In this section we shall show that the measures M converge to the a.c.i.m. . In Sect. 3 we shall study the rate of convergence. In Sect. 1, we considered the Perron–Frobenius operator. In our proof of Theorem 1 we need to modify this definition slightly. More generally, for any Hölder continuous function f V I ! R we can define the transfer operator Lf V C0.I / ! C0.I / by Lf k.x/ D X TyDx k.y/ef .y/: This operator Lf has a simple isolated maximal eigenvalue at the point eP .f /, where P.f / D P.f; T / D lim n!1 1 n log X x2Fix.n/ ef n.x/ is the pressure of f , and we denote f n.x/ D f .x/Cf .T x/C : : :Cf .T n−1x/[Bowen]. Given an analytic function g V I ! R and t 2 R we choose f .x/ D − log jT 0.x/j C tg.x/: The maximal eigenvalue of the corresponding transfer operatorL− log jT 0jCtg is therefore exp P.− log jT 0j C tg/. We define z.t/ to be its reciprocal z.t/ VD exp −P.− log jT 0j C tg/ : Note that setting t D 0 gives z.0/ D exp −P.− log jT 0j/ D 1, since 1 is the maximal eigenvalue of the Perron-Frobenius operator LT . We want to relate this to the integral of the function g with respect to the a.c.i.m.  via the following formula. Computing Invariant Densities and Metric Entropy 693 Lemma 5. Z g d D d dt P .− log jT 0j C tg/  tD0 : (see [Ru2, p. 133, Pa-Pol, p. 60]). In particular, since z.0/ D 1, Lemma 5 implies thatZ g d D − d dt log z.t/  tD0 D −z 0.0/ z.0/ D −z0.0/: .2:1/ We next want to relate the integral to the periodic points of T V I ! I . For a point x 2 Fix.n/ we denote r.x; n/ D 1 nj.T n/0.x/ − 1j : Summing over all such points we let an D an.t/ D X x2Fix.n/ r.x; n/etg n.x/; and then formally define the Fredholm determinant F.z; t/ by F.z; t/ D exp − 1X nD1 an.t/z n ! D exp 0 @− 1X nD1 zn n X x2Fix.n/ etg n.x/ j.T n/0.x/ − 1j 1 A : In Sect. 3 we will show that in fact F.z; t/ is an entire function of both z and t . For the moment let us assume this, and relate F to the expression (2.1) for R g d. The first step is the following lemma. Lemma 6. The function z.t/ VD exp −P.− log jT 0j C tg/ gives an implicit solution to F.z.t/; t/ D 0 satisfying z.0/ D 1. Proof. Note that z.t/ D exp −P.− log jT 0j C tg/ D lim supn!1 jan.t/j1=n is the radius of convergence of the power series P1 nD1 an.t/zn, and hence a zero of F (cf. [Ru2, p. 142], where similar statements hold for zeta functions). The fact that z.0/ D 1 was noted above. ut Now by the implicit function theorem we have that z0.0/ D −D2F.z.0/; 0/ D1F.z.0/; 0/ D −D2F.1; 0/ D1F.1; 0/ ; .2:2/ where DiF.z; t/, i D 1; 2, denotes the partial derivative of F with respect to the ith variable. Thus by (2.1) we can writeZ g d D z0.0/ D D2F.1; 0/ D1F.1; 0/ : .2:3/ We will now use the expression (2.3) to compute R g d. Both numerator and de- nominator can be expressed as series of rapidly decreasing terms, as follows. Using the expansion e−x D 1 CP1mD1.−1/m xmmW we can expand F.z; t/ D 1 C 1X ND1 CN.t/z N ; 694 M. Pollicott, O. Jenkinson where CN.t/ D X .n1;::: ;nm/ n1C:::CnmDN .−1/m mW an1.t/ : : : anm.t/: .2:4/ (Note the above summation is over all m-tuples of positive integers whose sum is equal to N ). With this expansion for F.z; t/ we can now calculate its partial derivatives, as follows. D1F.z; t/ D 1X ND1 NCN.t/z N−1; D2F.z; t/ D 1X ND1 C0N.t/zN : We can explicitly compute the derivative C0N.t/ as C0N.t/ D d dt 0 BB@ X .n1;::: ;nm/ n1C:::CnmDN .−1/m mW an1.t/ : : : anm.t/ 1 CCA D X .n1;::: ;nm/ n1C:::CnmDN .−1/m mW mX iD1 a0ni .t/ mY jD1 j 6Di anj .t/: Setting t D 0, and recalling the notation gni .x/ D g.x/Cg.T x/C : : :Cg.T ni−1x/, we obtain the following expression: C0N.0/ D X .n1;::: ;nm/ n1C:::CnmDN .−1/m mW mX iD1 X x2Fix.ni / r.x; ni/g ni .x/ mY jD1 j 6Di X z2Fix.nj / r.z; nj /: NowX x2Fix.ni / r.x; ni/g ni .x/ D X x2Fix.ni / 1 j.T ni /0.x/ − 1j gni .x/ ni D X x2Fix.ni / g.x/ j.T ni /0.x/ − 1j ; so that C0N.0/ D X .n1;::: ;nm/ n1C:::CnmDN .−1/m mW mX iD1 X x2Fix.ni / g.x/ j.T ni /0.x/ − 1j mY jD1 j 6Di X z2Fix.nj / r.z; nj / D Z g dN; where we define N D X .n1;::: ;nm/ n1C:::CnmDN .−1/m mW mX iD1 X x2Fix.ni / 0 B@ mY jD1 j 6Di X z2Fix.nj / r.z; nj / 1 CA xj.T ni /0.x/ − 1j : .2:5/ Computing Invariant Densities and Metric Entropy 695 The sum 1 C : : : C M is a finite signed measure, whose support is a union of periodic orbits. Since all points in a given orbit carry equal mass then 1 C : : : C M is T -invariant. Suitably normalized it gives a signed T -invariant probability measure M . This normalization is obtained by summing the first M coefficients in the power series expansion for D1F.z; 0/. Thus we define M D PM ND1 NPM ND1 NCN.0/ : .2:6/ Proposition 2. The measures M converge to the a.c.i.m.  in the weak topology. Proof. Let g.x/ D e2ikx . Then by (2.3), (2.5), (2.6) we see R e2ikxdM ! O.k/ as M ! 1. ut Remark 1. In general the signed measures M need not be positive measures. To see this it suffices to find some M such that 1 C : : : C M gives mass of opposite sign to two of its atoms. Suppose, for example, that T is symbolically the full shift on two symbols, with two fixed points x0 and x1, and a single period-two orbit fx01; x10g. Let ei D jT 0.xi/ − 1j−1 for i D 1; 2. Using (2.5) for N D 1 and N D 2 gives .1 C 2/.fx0g/ D e0 [.e0 C e1/ − 1] and .1 C 2/.fx1g/ D e1 [.e0 C e1/ − 1]. These weights are both of the same sign, but this sign might be either positive or negative depending on the derivative of T at x0 and x1. In particular we can find T with T 0.x0/ D 32 and T 0.x1/ D 3, say, in which case e0 C e1 D 52 > 1 so that .1 C 2/.fx0g/ > 0 and .1 C 2/.fx1g/ > 0. However the period-two point x01 is given zero weight by 1, and negative weight by 2, so that .1 C 2/.fx01g/ < 0 (the same is also true for x10). 3. Speed of Approximation So far in deriving our formulae, we have assumed that F.z; t/ is an entire function of both z and t . We now justify this. First recall (see [Gr]) that a bounded linear operator L V B ! B of a Banach space B over C is called nuclear if there exist normalised ui 2 B, normalised li 2 B, and i 2 C with Pi ji j < 1 such that Lw D 1X iD1 ili.w/ui: Lemma 7. F.z; t/ is an entire function of both z and t . Proof. For each piece Tai; aiC1U of the Markov partition we let Ti V I ! Tai; aiC1U denote the corresponding inverse branch of T (i.e. Ti  T D id on Tai; aiC1U), and zi 2 Tai; aiC1U the fixed point of Ti . For i D .i1; : : : ; in/, let Ti denote the composition Ti1  : : :  Tin , and write jij D n. For each such i, let zi 2 I denote the unique fixed point of Ti , and let Ii denote the image TiI . Let us choose n sufficiently large so that, for each i with jij D n, there is an open complex disc Di  Ii of radius di , centered at zi , such that 696 M. Pollicott, O. Jenkinson (a) for each 1  j  p, Tj extends holomorphically to the union D.n/ VD [jijDnDi , with Tj .D.n// a proper subset of Dj , (b) etg=jT 0j extends to a holomorphic function on each Di , and hence to a holomorphic function (which we denote by ) on the disjoint union Dn VD ‘jijDn Di . Let B denote the space of functions which are holomorphic on the disjoint union Dn and continuous on Dn. Equipped with the uniform norm, B is a Banach space. The operator L− log jT 0jCtg is well-defined on B, and takes the form L− log jT 0jCtgw.z/ D pX jD1 .Tj .z//w.Tj .z//: This operator is then nuclear (see [Ru1, p. 236]), and in particular the trace trace.Lm− log jT 0jCtg/ D 1X iD1 emi < 1 is well-defined for each m, where the ei are the eigenvalues ofL− log jT 0jCtg counted with multiplicity. By Fredholm theory [Gr] we have the trace formulae trace.Lm− log jT 0jCtg/ D X x2Fix.m/ etg m.x/ j1 − .T m/0.x/j ; and we can identify F.z; t/ D det.I − zL−