Physica 25D (1987) 387-398 North-Holland, Amsterdam
THE ENTROPY FUNCTION FOR CHARACTERISTIC EXPONENTS
Tomas BOHR Physics Lab. 1, H. C. Orsted Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen O, Denmark
David RAND * Department of Mathematics, University of Arizona, Tucson, A Z 85721, USA
Received 9 June 1986 Revised manuscript received 1 August 1986
Using a thermodynamic formalism, we define an entropy function S(a) which measures large deviations of the Liapunov characteristic exponents of certain hyperbolic dynamical systems. The function S(a) is a Legendre transform of a free-energy or pressure associated with the dynamical system. We show that S(a) is the noncompact topological entropy of the set of points A, with characteristic exponent a, that S(a)/a is the Hausdorff dimension of A~ and that a - S(a) is the escape rate from A,. We explain how to use the formalism for cookie-cutters to describe the distribution of scales in the universal period-doubled attractor and critical golden circle mapping. We relate S(a) to the Renyi entropies, prove a conjecture of Kantz and Grassberger relating the escape rate from hyperbolic repellors and saddles to the characteristic exponents and information dimension and study the fluctuations of escape rates. We discuss the escape rate from f(x) = (4 + e)x(1 - x) and the behaviour of the associated S(a) as e "~ 0. Finally we discuss how to apply these ideas to experimental time-series and non-hyperbolic attractors.
1. Introduction measure). We also obtain a formula for the escape rates from hyperbolic repellors and saddles and In this paper we use a thermodynamic for- prove a conjecture of Kantz and Grassberger re- malism to associate to a hyperbolic attractor, lating the escape rate to the information dimen- saddle or repellor, A, a real-analytic exponent sion and characteristic exponent. We then study entropy function, S(a), which measures the large the fluctuations of escape times and escape times fluctuations of characteristic exponents from the from other interesting sets and discuss the escape mean. It will turn out that S(a) is the topological rate from the invariant set of the quadratic family entropy of the set of points whose characteristic f(x) = (4 + e)x(1 - x) as e N 0. Finally, we dis- exponent is a, that S(a)/a is the Hausdorff di- cuss how to apply these ideas to experimental mension of this set restricted to an unstable mani- time-series. fold and that a - S(a) is the escape rate from this Many of the basic ideas are taken from Rand set. Also, from S(a) one can read off the various [20] where a more general treatment is developed entropies (topological, metric and Renyi) associ- including applications to the singularity or dimen- ated with A and its Sinai-Ruelle-Bowen measure. sion spectrum (Frisch and Parisi [9]; Benzi et al. We relate S(a) to an entropy function for the [1]; Halsey et al. [13]). Indeed, S(~) is directly natural or Sinai-Ruelle-Bowen measure (SRB related to the singularity spectrum f(a) for the measure of maximal entropy (see section 5). In turn, the ideas in [20] depend crucially upon the
* Permanent address: Mathematics Institute, University of thermodynamic formalism developed by Bowen, Warwick, Coventry CV4 7AL, UK. Lanford and Ruelle [2, 16, 22].
0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 388 T. Bohr and 1). Rand/Entropy function for characteristic exponents
For clarity we are not going to treat the general case here, but instead concentrate on the proto- typical example of Sullivan's 'cookie-cutter'
Cantor sets. In this way we avoid non-essential A1 technical problems and quickly get down to the I0 I~ heart of the problem. The extension to general ^2 hyperbolic attractors and more general hyperbolic saddles and repellors is straightforward. By revers- ing time, one can also obtain an analogous Fig. 1. Schematic representation of the cylinders of a cookie- entropy function which gives the Hausdorff di- cutter. mension of the set of points with backwards char- acteristic exponent equal to a in the intersection of a local stable manifold with A. To deduce these assume that If'l > 1. Let results one uses the ideas developed in this paper and the results about Markov partitions in [18]. A= {xeI: fJx~I forj=O,1,2 .... } These ideas should also apply to a large class of non-hyperbolic strange attractors with 1-dimen- and let h: A ~ Z = (0,1} ~ be defined as follows: sional unstable manifolds such as the Henon at- h(x) = a(O)a(1)a(2) .... where a(i) ~ (0,1} is tractor though there is likely to be some physically such that fix ~ Ia( o. Then it is easy to show that interesting modifications (see, for example, the h is a homeomorphism from A to Z (when 2: has map f0 discussed in section 10). There one uses the product topology) and (n,e)-separated sets ([2]) instead of cylinders but hof=ooh, otherwise the formalism but, of course, not the rigorous results goes through as described here for where o is the shift: o(a(O)a(1)a(2)...) = cookie-cutters. We explain a method to determine a(1)a(2)a(3) .... This symbolic representation will S(a) from experimental time-series in section 11. be useful for what follows although we will not This formalism for cookie-cutters can also be ap- use it explicitly. plied to the analysis of the spectrum of scales in Let (a) the attracting Cantor set for the fixed point of the period-doubting operator and (b) the golden A,=(x~I: f&EI fori=0,1 ..... n}. fixed point for critical maps of the circle as is explained below in section 2. The components of A, are called n-cylinders and the set of n-cylinders is denoted cg,. Clearly, the set of n-cylinders can be indexed by the elements 2. Cookie-cutter Cantor systems of (0,1}". We note here that our results still hold if we A cookie-cutter Cantor system is defined as relax the condition that If'l > 1 and just demand, follows. Let I = [0,1] and I 0, 11 c I be two dis- for example, that for some n> 1, I(f")'l > 1 joint closed subintervals. Let f: I 0 U 11 ~ R be a wherever f" is defined. An example of such a C 1+~ map* such that f(Io) = I and f(I1) = I. We system is given by the mapping fix) = (4 + e)x(1 - x), e > 0, where I 0 and 11 are the two compo- nents of f-l(I). The criterion is satisfied since f *A function q0 is HSlder continuous with exponent a if has negative Schwarzian derivative. In section 10 there exists c > 0 such that Iq0(x) - ep(y)l < clx -yl a for all we shall discuss some results about such systems, x and y. A map f is C 1+" if it is C 1 and its derivative f' is H/Slder continuous with exponent a. in particular the dependence of S(a) upon e. T. Bohr and D. Rand/Entropy function for characteristic exponents 389
Another interesting example* of a cookie-cutter there exists a constant P and constants Cl, ¢2 > 0 is given by the quadratic fixed point g--1- such that for all n ~ N, all C ~ ~ and all x ~ C, 1.527x 2 + 0.1048x 4 + 0.0267x 6 + • • • of the dou- bling operator for unimodal maps of the interval /%(C) ~ [c 1, c2]e -"p+s"~°(x), (1) (Feigenbaum [6], Coullet and Tresser [4], Lanford [17]). Let x o be the critical point of g and x, = where Snq~(x ) = cp( x ) + ... + ¢p( fn- lx ). It is easy g"(Xo). If a = 2g(xl) let to deduce from (1) that
a-Ix on I 0 = [x2, x4] , P = lim n -1 log ~ e s"~°(c), f(x) = a-lg(x) on 11 = Ix 3, Xl], then f(Io) =f(Ii) = [x2, Xl] and If'[ > 1 except where S,q~(C) is the maximum value taken by at x = x 1. Moreover, the n-cylinders are the 2 n S,~p(x) on C and P(~) = P is called the topologi- intervals whose end-points are x i and Xi+z,, i = cal pressure (or just pressure) of % (In fact, 1 ..... 2 n. Thus the Cantor set A for f is exactly because of the principle of bounded variation the attractor for g though, of course, the dyna- described below, in this expression one could re- mics of f[A and g[A are completely different. place S, cp(C) by any value of S, cp(x) taken on C.) Feigenbaum has introduced a scaling function to The existence and uniqueness of #~o is due to study the various scales in this attractor [7]. It is Ruelle and Sinai and a proof is given in Bowen easy to see from this construction that the distri- [2]. To deduce the result as stated here one has bution of these scales is given by the function simply to apply the following principle of bounded S(a) for f. A similar construction can be done for variation. the golden fixed point of the renormalisation transformation on circle maps introduced in [8] Principle of bounded variation. This will play a and [19]. In fact, if (~, ~/) is the fixed point ([19]), crucial role in many arguments in the rest of the x o = 0, x 1 = ~(Xo), x~ = ~/(Xo), x 2 = ~T/(x0), a = paper. In fact when this fails one gets some physi- x~ - x 2 and I = [x~, xl] for f take cally interesting deviations from our results for a-ix ifx~Io=[X~,X2] , cookie-cutters, one example of which is described f(x) = in section 10. Suppose rp: I ~ R is H~51der con- a-l~l(x) if x ~ 11 = Ix2, Xl]. tinuous. Then, since the length of the cylinders is Note that in this case f(Io)= I, but f(I1)= Iv- exponentially decreasing with n, the variation of cp Consequently, the only symbol sequences on an n-cylinder decreases exponentially, i.e. there a(O)a(1)a(2)...that can occur are those without exists 0 0 such that for all n >_ 0 if two adjacent ls. Moreover, I 0 u 11 = I so that the C~ (~n then [cp(x)-cp(y)[
0.50 P(cp)=h(~)+ f~d~. (2)
This characterisation is known as the variational 0.25 principle for Gibbs states. It is due to Ruelle and Waiters and is proved in [2] (Theorem 1.22).
0 We shall be particularly interested in the Gibbs 1.2 Oh Ot2 2.0 state of - log If'[. This is the Sinai-Ruelle-Bowen Fig. 2. S(a) calculated numerically from P(fl) for f(x)= measure (denoted ~tSRB) which for Axiom A at- 4(1 + e)x(1 -x) with e = 0.5. P(fl) was calculated by taking tractors (with f' replaced by the derivative of f ratios of the sums )ZJk(C) ~ for the n-cylinders with n = 13 and along the unstable manifolds) is the natural mea- n = 14. The thin vertical lines mark limiting characteristic exponents a 1 = log(2 + 4e) = log(slope of f at positive fixed sure. Repellors and saddles do not possess a natu- point) and a 2 = log(4(1 + E)) = log(slope if f at origin). ral measure but the SRB measure is important in these cases because of its relation to escape rates and for a ~ R, as described below. We shall also use extens!vely the pressure P(fl) and Gibbs state /t• of ~ = S(a)=inf{S(S): aeJ). - fl log If'l- The function P(fl) is analytic ([22]). (Fig. 4 shows P(fl) for a cookie-cutter and fig. 6 (These definitions are motivated by large fluctua- shows it for a map, described in section 10, which tion theory for thermodynamics, particularly is not a cookie-cutter and where the behaviour of Lanford's paper [16].) P is very different.) Now let R denote the set of limit points of the sequences n- 1S.~ (x) where x ranges over A. Then R is an interval (see [20], section 4). Clearly out- 4. Entropy function for Liapounov exponents. side R, S(a) = - oo. Moreover, by methods simi- lar to those used in [16] it is easy to prove that We now introduce the entropy function. Given S(a) is continuous and concave on R. An exam- a cookie-cutter f we let ple is shown in fig. 2, and for a system which is not a cookie-cutter in fig. 5. n-I Now recall the definition of the pressure P(fl) Xn(X) =rl-1 E loglf'(fJx)] = -n-ls.(~(x), of -fl log If'[ given above. Provided ¢p = j=O - log lf'l is not of the form c + u o f- u where c where ~ = -log If'l, so that if the limit exists, is a constant and u a Htlder continuous function, X(X) = lim,~ X,(x) is the characteristic expo- then P(fl) is a strictly convex function of fl [22] nent associated with the orbit of x. Now let J be and R is a non-trivial interval [2]. We shall as- an open interval and let N,(J) denote the number sume this to be the case. [Otherwise, P(fl) is of cylinders C in W, such that X,(x) ~ J for some affine and there is no variation in the characteris- x in C. Let tic exponents.] Then the map dP S(J)= lira n-llogN.(J) 13 ~ a - dr (13) = txB(log If'l), i1---~ O0 T. Bohr and D, Rand/Entropy function for characteristic exponents 391 which sends/3 ~ R to R, has an analytic universe (4) Consider the value of a where S'= ft = 1. ft = ft(a). [We henceforth use the notation /~(q~) Then for f+ d#.] S(a) = P(1) + a(1) Theorem 1. S(a) = P(fl(a)) + aft(a), i.e. S and P = P(-log If'l) - If'l)d#SRB are Legendre transforms of each other. f(-log = hSRB, Proof. Let J = [a, b] be an interval containing a, ft = ft(a), P = P(ft), cp = -log [f'[ and m, be the the metric entropy of the SRB measure. The last /t a -measure of the union of the n-cylinders C equality follows from the variational principle for such that -X,(X) ~ J for some x ~ C. Then, since Gibbs states. /~a is the Gibbs state of ftcp with pressure P and (5) The Hausdorff dimension HD of A is given using the bounded variation principle, there exist by the value of/3 such that P(ft)= 0 (Manning constants dl, d 2 > 0 such that and McClusky [18]). In our case this is easily seen since, if ~ denotes Lebesgue measure (i.e. length), m. ~ [dr, d2] ~ e -"P+os"~(~c), the ratio of
Y'~X(C) a and ~ e as"~(c) where x c ~ C and the sum is over all C ~ c¢, such that -X~(x) ~ J for some x in C. Thus, if/3 > 0, is uniformly bounded from above and below (use d,N~([a, b]) e --(e+~b)~ the bounded variation principle). Hence they have < m~ < d2N~([a, b])e -(p+aa)n. the same growth rates P(ft). Consequently, setting /3 = HD we get
The reverse inequalities hold if ft _< O. But m, ~ 1 S(a(HD)) = HDa(HD) as n-, o¢ because [a, b] contains the #a-mean a(ft) of log If '1, so the result follows by taking the = S'(HD) a(HD) limits n ~ oo and a, b ~ a. [] since /3 = S'. Consequently, HD is the value of S(a)/a for which (S(a)/a)'= O. This statement We note the following direct corollaries of this is generalized below. result: (6) It follows directly from Theorem 1 that (1) S(a) is real-analytic on R. (2) S'(a) = ft. In particular S' takes all values s,,(a) = - 1/P,,(ft(a)) between _ oo. (3) Since S is concave and S' takes all values, S and from the thermodynamic formalism of Ruelle has a unique critical point a c which must be a that maximum. At ac, /3 --- 0 by 2 so, by Theorem 1, P"(fl) =m a = lim n -1 fl S, cp- n/ a(eP) [2 d#a' S(ac) = P(0) = h,op n -.--~ oO (3) is the topological entropy. This is the metric en- tropy of the so-called measure of maximal entropy where ¢p = -/3 log If'l- Thus which is obtained by giving all the cylinders equal weights. S"(a) = - 1/ma(,). 392 T. Bohr and D. Rand/Entropy function for characteristic exponents
We shall use this later to study fluctuations. This Y. Let second moment has been studied before by Grassberger and Procaccia [12] and Fujisaka [10]. h~(Y) =inf (t: m~,t(Y ) =0). (7) It follows from (5) that S is an invariant of bi-Lipshitz conjugacy i.e. if h o f= g o h with h a Then the topological entropy h(Y) of Y is Lipshitz homeomorphism with a Lipshitz inverse sup~ h~(Y). In [3] it is proved (a) that this defini- then the S(a) curves for f and g are identical. tion agrees with the usual one when Y is compact, This follows because if C is a cylinder for f and (b) that if v is an invariant Borel probability h(C) the corresponding cylinder for g then measure and v(Y) = 1 then h(Y) > h(v) where k-iX(C) < ~( h(C)) <_ k~(C) provided k is larger h(v) is the metric entropy of v and (c) that if Y is than the Lipshitz constants for h and h -1. Thus the set of generic points of the ergodic invariant measure v then the functions P(fl) for f and g are identical. h(Y)= h(v).
Note. Many of the results in this section did not Proof of Theorem. If fl = fl(a), then using Theo- rem 1 and the variational principle for Gibbs depend upon the fact that ~o = -log [f'l and are equally valid for any HiSlder continuous function states, cp: I~R. = +
= P(3) - I~B(fl log If'l) -- h (~w),
the metric entropy of/~. Thus since/~(A ~) = 1, 5. The meaning of S(a) and D(a) -= S(a)/a h(A~) < h(~ta) = S(a). Now we prove the reverse inequality. If J is an interval containing a in its Let A~ denote the set of points x for which interior let ,In = ( x ~ A: Xm(X) ~ J for all m > n }. lim. ~ oo exists and equals a. Clearly this set Xn(X) Let c~,,(j) be the set of all those m-cylinders C is invariant under f though in general it is not such that Xm(X) ~J for some x in C. Then c~,~(j) compact. is a cover of J, for all m > n and if ~ = gk is the cover of A by k-cylinders, k < n, Theorem 2. S(a) and D(a) = S(a)/a are respec- tively the noncompact topological entropy and m~,,(J,) < lim E e-'(~-k) m-,oo ~¢,.(j) Hausdorff dimension of A ~. = lim Nm(J)e -'(m-k). We must first recall Bowen's definition of topo- logical entropy for noncompact sets [3]. Then we Thus, if t > S(J) then m~,t(J,) = 0 i.e. ha(J,) < give the proof. Let f: I ~ I be given and suppose S(J). Since this is true for all k _> 0, it follows that that Y c 1. Let ~ be a finite open cover of I. If E h(J,) < S(J). Consequently, since A~ c U,Jn, is a subset of 1 let n(E) denote the smallest h( A ~) < hO,J ,J,) = sup, h( J,) < S( J). Taking the non-negative integer such that f"(e)+l(E) is not limit J ",~ a gives the required result. contained in a single element of ~. An e-cover of Now consider D(a)= S(a)/a. It follows from Y is an open cover g such that e -"(E) < e for all the results of Young [23] that if/~a(~)(E) = 1, then E ~ g. Define the Hausdorff dimension HD(E) of E is not less than h(Ixt~(~))/a = D(a). Thus HD(A~) >_ D(a). m~,t(Y)= lim inf (~ e-t~CE)), E~O ~ ,y Let .,~ffa(E) = inf E X(B), where the infimum is taken over all e-covers g of BE~ T. Bohr and D. Rand/Entropy function for characteristic exponents 393 where the infimum is over all covers ~ of E by or. But intervals such that X(B) < 8 for all B ~ ~ and let .Vt°a(E) = sups>0.gt°a(E) = lims_.0.,~d(E). Let a(x)= lim log~,(C,,x)/log)~(C,.x), n ----* oo J be an interval containing a in its interior. If ?~(C) denotes the length of C then by the princi- where C,, x is the n-cylinder containing x. But ple of bounded variation and the definition of v(C,,x) = 2-" and, if the limit exists, cg,(j), there exists constants c x, c2> 0 indepen- lim,~o~n llogX(C,,x)=-X(X) so f(a) is the dent of n such that Hausdorff dimension of the set of points with characteristic exponent a- 1 log 2, i.e. D(a- 1 log 2). ~_, X(C)a E [c,,c2]N,(J)e -"aJ. ~.(J) 6. The metric exponent entropy function Thus, since c~,,,(j) is a cover of J, for m > n, if We now briefly discuss a new entropy function J = [a, b], a(a) which is defined in terms of the SRB measure rather than the cylinders. We will use this in our .;¢~,d( j~) < c2N,.( j ) e-,,aa, study of escape rate fluctuations. If J is an open interval let provided m is chosen so large that 8> e -ma. Consequently, if d>S(J)/J then ,,~d(j,)= o(J)= lim n-'log/~sR.{x:x.(x)~J} lims~0.,~d(J,) = 0, whence .,~d(U,J,) = 0. But n -"* O0 A ~ c U,J,, so d > implies S(J)/J a'g'd(A~) = O. and Taking the limit J xa a one deduces that if d > S(a)/a then .~d(Aa) = 0, whence HD(A~) < o(a)=inf{o(J): a~J}. D( a) = S( et)/a. [] Theorem 3. o(a) = S(a) - P(1) - a.
In particular, S(a(O)) is the topological entropy Note. It follows from this that S(a) - P(1) - a _< and D(a(1)) is the information dimension. The 0 with equality if and only if a = a(1) the #SRB- last results is true because, by the above, D(a(1)) mean exponent. = hsRa/X, where X=ffsRa(log If'l) is the /ZSRB- average characteristic exponent, and, by the re- Proof. Let s(J) and s(a) be defined exactly as suits of Young implicit in [23] (a direct proof for o(J) and a(a) except that the set {x: X,(X) ~J} this simple case is given in ref. 20 (Theorem 5 and is replaced by the union Cj of all n-cylinders C Corollary), h SRB/X is the information dimension. such that X,(X) ~ J for some x in C. Then if d is This last relation has been conjectured by the constant given by the principle of bounded Grassberger in [11]. variation of a ~ (a, b), We note from the above that S(a) is related to the singularity spectrum f(a) [13, 20] of the mea- s([a + 2d/n, b- 2d/n]) < a([a, b])
where P = P(1). Thus ~tsp.B(Cj) is in the interval principle. Consequently, there exist d > 0 such that if S,q~(C)= maxx~cS,~p(x ) then N,(J)e -"P-"J. E < Ex(c) -< E e s°,(c)+" Letting n ~ oo and I ",~ a we get s(a)= S(a)- P(1) - a and hence the result. [] and therefore ER =-P(~). Applying the varia- tional principle for Gibbs states (2) to this, we obtain the following result conjectured by Kantz 7. Escape rates and Grassberger [15] and discussed by Eckmann and Ruelle in [5]. For e > 0, let
Theorem 4. ER = X - hSRB = X( 1 - HD(/~SRB)). a,(e)= {x~I: d(fJx, A)
U Uc. But, using the principle of bounded variation, ~¢. +k ¢¢" one can deduce that ?~(A,(A)) has the same growth rate as E~(C) where the sum is over all The results follow directly from this. [] n-cylinders C such that X,(X) ~ A for some x ~ C. But there are N,(A) terms in this sum and there Now we explain why ER = -P(-log If'l)- This are positive constants c 1 and c 2 (independent of n result is due to Bowen and Ruelle for general and C) such that each term is contained in the hyperbolic sets and agrees with the calculation of interval [ cl, c 2 ]e-"a. Consequently, the growth rate Kadanoff and Tang [14]. We have is contained in the interval S(A)- A and letting A "~ a one deduces the following result. EX(C) = E exp(-l°gl(f")'(xc)l) Theorem 5. a- S(a) is the escape rate ER(a) for some xc~C. Since ¢p=-loglf'l is H61der from the set of points A~ with characteristic expo- continuous, we can apply the bounded variation nent a. T. Bohr and D. Rand~Entropyfunction for characteristic exponents 395
8. Fluctuations in the escape rate Theorem 6. h q = (1 - q)-l( p( q) _ qP(1)).
We shall be interested in a specific small Proof. If tp = -log If'l and P = P(1), fluctuation c from the /~sRB-generic escape rate ER = a(1) - S(a(1)). This is achieved by (1 -- q)hq = lim n-' log E~SRB(C) q
(a - a(1)) = 2,: + 0(c3). = lim n -1 log ~ e -nPq-qS.w(C) n-'~ (ban At the n th level the proportion of points x with = (P(q) - qe(1)). [] escape rate approximately corresponding to a (re- place a by an interval to make this precise) is Thus, using Theorem 1, one can read off all the Ex(c)/Zx(c) hq from S(a) and conversely, if one knows P(1) (using Theorem 3 for example), which one also needs in order to measure S(a) by section 6. In = y~. eS.~(x~)-"P/y" eS.,(xc)-,e, general, it is likely to be easier to measure P(q) or hq and to determine S(a) from this by using Theorem 1. where the first sum is again over those n-cylinders which contain a point x such that X,(x)=a, ¢p = -log If'l and x c is some point in C. Conse- 10. Quadratic maps of the interval quently, this has the same decay rate as
As we explained above one can regard the map I*l~{x~I: X.(x)=a}, f~(x) = 4(1 + e)x(1-x) as a cookie-cutter when e > 0. We measured the escape rates for this as which is o(a). But, for small c, we have e ~ 0 and it appears that these scale like e 1/2. Moreover, the corresponding entropy functions o(a)=(1/2)o"(a)(a-a(1)) 2 +O((a-a(1)) 3) S~(a) have an interesting behaviour as e x~ 0. They converge to S O which is calculated numerically in = _ (2/ml)c 2 + 0(c3), fig. 5. This should be compared with the graph of So ooa which is plotted in fig. 3. Notice that So(a ) where m 1 is given by (3). Thus small fluctuations c in the escape rate decay like exp (-2ncZ/ml). 0.75 / ,/I 9. Generalised entropies 0.50
This definition is due to Renyi [21]. The q- entropy for/, sRa is defined to be 0.25 1 hq 1 - q lim n -a log 2~SRB(C) q. n~oo ~b~n 0 0.5 (xl ct2 1.5 It is easy to check that h 0 = hto p and limq..~ 1 hq = h SRB. Fig. 3. As fig. 2 except e = 0.001. The diagonal line is y = a. 396 T. Bohr and D. Rand/Entropy function for characteristic exponents
3C I 4 l Presumably the apparent nonlinearity of the graph is due to the fact that large n behaviour is not well approximated by n as small as 13. Con- sider now the behaviour of the pressure function 10 P(fl) (see fig. 6). Here an n-cylinder is a maximal closed interval on which f" is monotone and we 0 define P(fl) to be the growth rate of Z?~(C) a where the sum is over all n-cylinders (see [2] for -1(3 L-- the general definition of pressure for arbitrary dynamical systems). The homeomorphism g: -20 -- [0, 1] 3 given by g(x) = sin 2 (xTr/2) conjugates J I I -20 -10 0 I0 20 30 F(y)=l-[2y-l[ to f0 i.e. foog=goF. The homeomorphism g fails to be a diffeomorphism Fig. 4. P(fl) for the f as in fig. 3 calculated numerically by taking ratios of the sums E~,(C) ~ for the n-cylinders with because g' = 0 at x = 0,1 so that its inverse is not n=13and n=14. even Lipshitz there. Now, by the mean-value theo- rem for each C there exists x c ~ C such that does not qualify as the entropy function of a cookie-cutter even though it is concave and ana- lytic. For example, S' does not take all values. EX(C)~ = E2-"Bg'(xc) ~ The map f0 is not a cookie-cutter because, al- = 2"(1-~)~2-"g'(xc) ~. though one can define cylinders in the obvious way, log If'l is not bounded below. This affects our analysis mainly through the failure of the Thus if the last sum is bounded away from 0 and bounded variation principle. Because one can con- then E?~(C) a has growth rate (1 -/3)1og2. But jugate this to a piecewise-linear map, one can see this sum is estimated by fg'(x)/~dx and the only that the behaviour is dominated by n-cylinders way that this integral can fail to be bounded away whose length decays as 2-". In fact fig. 5 is very from 0 and oo is because of the contributions from misleading because S(a)=21og2-a, i.e. a neighbourhoods of 0 and 1 which can cause it to straight line, as we shall now show. diverge. But ifx=eor l-e, g(x)-e 2 as e~0
0.75 300 / 0.5(
0.0 0.2! I
0.0( - 20.0 0.5 al a a 2 1.5 -20 0.0 25
Fig. 5. As fig. 3 except that e ~ 0. Fig. 6. As fig. 4 except that ~o = 0. 7". Bohr and D. Rand/Entropy function for characteristic exponents 397 so g'(x) - e. Thus the integral is bounded exactly case the natural measure is induced by the map F when fl > - 1. From this we deduce that P(fl) = and that this measure is equal for all cylinders. (1-fl)log2 when r>-1. [One can easily see Thus the singularity spectrum and dimensions Oq that P(fl) must be of this form for fl>0 by of [13] and [20] can be read off by the formula of noting that since P'(fl)~-log2 as fl-~ o¢ section 5. In particular, Dq= 1 for q<2 and and since P(fl) is convex, P'(fl)<-log2 for Dq = (q - 1)-1/2 for q >_ 2. all ft. But P(0)=log2 and P(1)= 0 (since the Hausdorff dimension is 1) whence the slope must be exactly -log2 for all fl > 0.] Now we show 11. Experimental time-series that for all fl_< -1, P(fl) = -2fllog2. First we note that for all r, P(fl) > -2fl log2 because 0 is Finally a word about how to use these ideas to a fixed point and f'(0) -- 4. Thus if Cend is one of measure experimental strange attractors. The idea the two end n-cylinders then h(Cend)- 4 -n as is to construct cylinders by approximating the n ~ o¢. Thus since EX(C) a >_ h(Cend), P(fl) > strange attractor by a symbolic dynamical system. -fl log4. It remains to show that if fl < -1 then To do this one should partition the phase space up P(fl)_<. 2r log2. To see this we break the sum into sensibly chosen regions labelled, for example, over all cylinders up into three sums S 1, S 2 and S 3 1 ..... m. Then one can code the experimentally which are respectively over the two end cylinders, measured orbit of a Poincar6 map by the sequence those cylinders which intersect [e, 1- e], and the of regions it visits. This gives a symbolic orbit remainder. Let us deal with S 2 first. We chose which is a sequence of elements of (1 ..... m }. The e so that on the interval [e,l-e], g'>e/2. set of sequences of length n which occur in this Then 3 2 = ~..gt(Xc) B " 2 -nt~ < E(e/2) ¢ • 2 -he < orbit represent the n-cylinders. Let o 1..... o,~ de- 2n.(e/2)¢.2 -he, where the sums are over the note the distinct cylinders that occur and let x i, appropriate set of cylinders. Thus the growth rate i = 1 ..... m, denote a point in phase space whose of S 2 is <(1-fl)log2 which is <-2fllog2 if initial symbol sequence is o i. Then one can ap- fl < - 1. Now for S 3 one has S 3 = Eg'(xc) t~. 2 nO proximate P(fl) by Zi exp ( - flnxn(xi) ). From this < 2cE(j. 2-n)B2 -he, where the constant c is in- one can calculate S(a) using the formula of Theo- dependent of n, the sum is over the appropriate rem 1. Moreover, to fix an unstable manifold set of cylinders and j runs from 1 to the smallest approximately, choose a large integer N and a integer greater than e nG. To deduce this we have sequence X in (0 ..... m} u. Now scan the full used the fact that on the interval [0, e], g'(x) > cx symbolic orbit and every time the sequence X for some c > 0 and that j-2-" is the left-hand occurs note the sequence Y of length n which endpoint of the (j + 1)st cylinder from the left. occurs just before it. These give the n-cylinders Thus S 3 < 4-n¢Ej ¢ and this last sum is conver- restricted to this approximate unstable manifold. gent when fl < - 1 so that the growth rate of S 3 is < - 2fl log 2. We have already seen that the growth rate of S 1 is exactly -2fl log2 so we deduce that Acknowledgements P(fl) = -2fl log2 when fl < -1. Thus the graph of P(fl) consists of two affine pieces with slopes We are extremely grateful to Oscar Lanford for - log 2 and - 2 log 2 which meet at fl -- - 1. (In pointing Rand to his paper [16], to Preben the thermodynamic analogy there is a phase tran- Alstrom, Anthony Manning, Felics Przytycki and sition at fl=-1.) The claimed form of S(et) Peter Walters for several useful conversations, to follows from this. Such behaviour is very non- Robert MacKay for his criticisms of a previous generic and dependent upon the existence of the draft and to the referee for a number of interest- conjugacy g. It is interesting to note that in this ing comments. This paper was written during a 398 T. Bohr and D. Rand~Entropy function for characteristic exponents visit to the Special Year on Chaos and Turbulence [10] H. Fujisaka, Statistical dynamics generated by fluctua- at the University of Arizona which was partially tions of local Lyapunov exponents, Progr. Theoret. Phys. 70 (1983) 1264. supported by the US ONR under grant N00014- [11] P. Grassberger, Information flow and maximum entropy 85-K-0412 and by the University of Arizona. It is measures for 1-D maps, Physica 14D (1985) 365. a great pleasure to thank the University of Arizona [12] P. Grassberger and I. Procaccia, Dimensions and entro- pies of strange attractors from a fluctuating dynamics Mathematics Department and Applied Math Pro- approach, Physica 13D (1984) 34. gram for their hospitality and Andrea Morrison [13] T. Halsey, M. Jensen, L. Kadanoff, I. Procaccia and B. for typing the manuscript. Shraiman, Fractal measures and their singularities: the characterisation of strange sets, preprint (1985). References [14] L. P. Kadanoff and C. Tang, Escape from strange repel- lors, Proc. Nat. Acad. Sci. USA 81 (1984) 1276. [15] H. Kantz and P. Grassberger, Repellors and semi-attrac- [1] R. Benzi, G. Paladin, G. Parisi and A. Vulpiani, On the tors/md long-lived chaotic transients, Physica 17D (1985) multifractal nature of turbulence and chaotic systems, J. 75. Phys. A 17 (1984) 3521. [16] O. Lanford, Entropy and equilibrium states in classical [2] R. 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