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The Entropy Function for Characteristic Exponents 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.
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