10 Circle Maps: Irrationally Winding - Center for Nonlinear Science

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20 P. Cvitanovic q 0 max = ln j 1j (42) = 1:502642 ::: ln 2 However, for large and negative, q() is dominated by the interval (15) which shrinks only harmonically, andq() approaches 0as (43) q() = 3lnn n ln 2 ! 0: So for nite n, q n () crosses the axis at ; = D n , but in the n ! 1 limit, the q() function exhibits a phase transition q() = 0 for < ;D H , but is a non-trivial function of for ;D H . This non-analyticity is rather severe - to get a clearer picture, we illustrate it by a few number-theoretic models (the critical circle maps case is qualitatively the same). An cute version of the \trivial" Farey level thermodynamics is given by the \Farey model"[19], in which the intervals `P=Q are replaced by Q ;2 : (44) Z n () = 2 n X i=1 Q 2 i : Here Q i is the denominator of the ith Farey rational P i =Q i . For example (see (denition (10.3)), Z 2 (1=2) = 4 + 5 + 5 + 4: Though it might seem to have been pulled out of a hat, the Farey model is as sensible description of the distribution of rationals as the periodic orbit expansion (34). By the \anihilation" property of the Gauss shift (11), the nth Farey level sum Z n (;1) can be written as the integral (45) Z n (;1) = Z dx(f n (x)) = X 1=jf 0 a 1 :::a k (0)j with the sum restricted to the Farey level P a 1 +:::+a k =n+2 . It is easily checked that fa 0 1 :::a k (0) = (;1) k Q 2 [a 1 :::a k ],sotheFarey model sum is a partition generated by the Gauss map preimages of x = 0, ie. by rationals, rather than by the quadratic irrationals as in (34). The sums are generated by the same transfer operator, so the eigenvalue spectrum should be the same as for the periodic orbit expansion, but in this variant of the nite level sums we can can evaluate q() exactly for = k=2, k a nonnegative integer. First one observes that Z n (0) = 2 n . It is also easy to check that[27] Z n (1=2) = P i Q i = 2 3 n . More surprisingly, Z n (3=2) = P i Q 3 = 54 7 n;1 . Such \sum rules", listed in the table 10.1, are consequence of the fact that the denominators on a given level are Farey sums of denominators on preceding levels[63, 19]. Regretably, we have not been able to extend this method to evaluating q(;1=2), or to real . A bound on D H can be obtained by approximating (44) by (46) Z n () = n 2 + 2 n 2n : In this approximation we have replaced all `P=Q , except the widest interval `1=n , by the narrowest interval `Fn;1=F n (see (16)). The crossover from the harmonic
10. Circle Maps 21 dominated to the golden mean dominated behavior occurs at the value for which the two terms in (46) contribute equally: D n = ^D + O ln n ! ln 2 ^D = = :72 ::: n 2ln (47) For negative the sum (46) is the lower bound on the sum (30) , so ^D is a lower bound on D H . The size of the level-dependent correction in (47) is ominous the nite n estimates converge to the asymptotic value logarithmically.Whatthis means is that the convergence is excruciatingly slow and cannot be overcome by any amount of brute computation. 2 q( ) Z n () = 0 2 2 Z n;1 1/2 3 3 Z n;1 1 (5 + p 17)=2 5Z n;1 ; 2Z n;2 3/2 7 7 Z n;1 2 (11 + p 113)=2 10Z n;1 + 9Z n;2 ; 2Z n;3 5/2 7+4 p 6 14Z n;1 + 47Z n;2 3 26:20249 ::: 20Z n;1 + 161Z n;2 + 40Z n;3 ; Z n;4 7/2 41:0183 ::: 29Z n;1 + 485Z n;2 + 327Z n;3 n/2 n = golden mean Table 10.1 Recursion relations for the Farey model partition sums (44) for = 1 1=2 1:::7=2 they relate the 2 q( ) = lim n!1 Z n+1 ()=Z n () to roots of polynomial equations. 10.13 Artuso Model The Farey model (30) is dicult to control at the phase transition, but considerable insight into the nature of this non-analyticity can be gained by the following factorization approximation. Speaking very roughly, the stability (;1) n Q 2 of a P=Q =[a 1 :::a n ] cycle gains a hyperbolic golden-mean factor ; 2 for each bounce in the central part of the Farey map (10), and a power-law factor for every a k bounces in the neighborhood of the marginal xed point x 0 = 0. This leads to an estimate of Q in P=Q = [a 1 :::a n ] as a product of the continued fraction entries[48] Q n a 1 a 2 a n In this approximation the cycle weights factorize, a1 a 2 :::a n = a1 a2 an ,and the curvature corrections in the cycle expansion (38) vanish exactly:
Page 1 and 2: 10 Circle Maps: Irrationally Windin
Page 3 and 4: 10. Circle Maps 3 10.1 Mode Locking
Page 5 and 6: 10. Circle Maps 5 10.2 Farey Series
Page 7 and 8: 10. Circle Maps 7 (7) x =[a 1 a 2 a
Page 9 and 10: 10. Circle Maps 9 (11) f a (x) = f
Page 11 and 12: 10. Circle Maps 11 Q r When the rep
Page 13 and 14: 10. Circle Maps 13 10.7 Global Theo
Page 15 and 16: 10. Circle Maps 15 Our understandin
Page 17 and 18: 10. Circle Maps 17 that the leading
Page 19: 10. Circle Maps 19 at the phase tra
Page 23 and 24: 10. Circle Maps 23 formally identic
Page 25 and 26: 10. Circle Maps 25 [29] P. Cvitanov

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