10 Circle Maps: Irrationally Winding - Center for Nonlinear Science

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12 P. Cvitanovic 10.6 Global Theory: Ergodic Averaging So far we have discussed the results of the renormalization theory for isolated irrational winding numbers. Though the local theory has been tested experimentally[40, 41], the golden-mean universality utilizes only a few of the available mode-locked intervals, and from the experimental pointofviewitwould be preferable to test universal properties which are global in the sense of pertaining to a range of winding numbers. We rst briey review some of the attempts to derive such predictions using ideas from the ergodic number theory, and then turn to the predictions based on the thermodynamic foramlism. The ergodic number theory[25, 42] is rich in (so far unfullled) promise for the mode-locking problem. For example, while the Gauss shift (6) invariant measure (19) (x) = 1 1 ln 2 1+x was known already to Gauss, the corresponding invariant measure for the critical circle maps renormalization operator R has so far eluded description. It lies on a fractal set - computer sketches are given in refs. [43, 44] - and a general picture of what the \strange repeller" (in the space of limit functions for the renormalization operator (12)) might look like is given in refs. [45]. Rand et al.[10] have advocated ergodic explorations of this attractor, by sequences of renormalizations R ak corresponding to the digits of the continued fraction expansion of a \normal" winding number W = [a 1 a 2 a 3 :::]. A numerical implementation of this proposal [43, 44] by Monte Carlo generated strings a 1 a 2 a 3 ::: yields estimates of \mean" scalings =15:5 :5 and =1:8 :1. n is the estimate of the mean width of an \average" mode-locked interval Pn=Qn , where P n =Q n is the nth continued fraction approximation to a normal winding number W =[a 1 a 2 a 3 :::]. In this connection the following beautiful result of the ergodic number theory is suggestive: (10.1) Theorem (Khinchin, Kuzmin, Levy[42]) For almost all W 2 [0 1] the denominator Q n of the n-th continued fraction approximant W = P n =Q n + n , P n =Q n =[a 1 a 2 a 3 :::a n ] converges asymptotically to (20) 1 lim n!1 n ln Q n = 2 12 ln 2 : In physics this theorem pops up in various guises for example, 2 =6ln2 can be interpreted as the Kolmogorov entropy of \mixmaster" cosmologies[46]. In the present context this theorem has been used[44] to connect the ergodic estimate of to ^ estimated[5]byaveraging over all available mode-lockings up to given cycle length Q, but it is hard to tell what to make out of such results. The numerical convergence of ergodic averages is slow, if not outright hopeless, so we abandon henceforth the ergodic \time" averages (here the \time" is the length of a continued fraction) and turn instead to the \thermodynamic" averages (averages over all \congurations", here all mode lockings on a given level of a resolution hierarchy).
10. Circle Maps 13 10.7 Global Theory: Thermodynamic Averaging Consider the following average over mode-locking intervals (4): (21) () = 1X X Q=1 (P jQ)=1 ; P=Q : The sum is over all irreducible rationals P=Q P
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Page 9 and 10: 10. Circle Maps 9 (11) f a (x) = f
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10 Circle Maps: Irrationally Winding - Center for Nonlinear Science

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