F. Christiansen, G. Paladin and H.H. Rugh, Nordita preprint (Jan 1990), submitted to Phys. Rev. Lett..  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the Jensen et al. dimension manifest. As a practical formula for evaluating this dimension, 39) has so far yielded estimates of DH of modest accuracy, but that can surely be improved. In particular, computations based on the (34) infinite products should be considerably more convergent[53, 54], but have not been carried out so far. The derivation of (39) relies only on the following aspects of the hyperbolicity conjecture of refs. 29, 45, 55, 32] 1) limits for Shenker ffi s exist and are universal. This should follow from the renormalization theory developed in refs. 9, 10, 33] ....

....ref. 8] the explicit values of the asymptotic exponents and prefactors were not used, only the assumption that the growth of ffi p with n is not slower than a power of n. Explicit evaluation of the spectrum was first attempted in ref. 23] prerequsite for attaining the exponential (or faster[53, 54]) convergence of the cycle expansions are effective methods for summation of infinite families of modelockings. At present, those are lacking none of the tricks from the Riemannzeta function theory (integral representations, saddle point expansions, Poisson resummations, etc. have not worked for ....

F. Christiansen, G. Paladin and H.H. Rugh, Nordita preprint (Jan 1990), submitted to Phys. Rev. Lett..

.... irrational windings set of the critical circle maps[13] to the Hamiltonian period doubling repeller[14] to a Hamiltonian three disk pinball[15] to the three disk quantum scattering resonances[16, 17, 18] to the non leading eigenvalues[19] of transfer operators and to the correlation exponents[20]. Feasibility of analysis of experimental strange sets in terms of cycle expansions is discussed in refs. 6, 21] Acknowledgements: These lectures are a presentation of extended recycling project[7, 8, 19] with R. Artuso and E. Aurell. The derivation of the Selberg product of sect. 2 is based on ....

....i 0 = i 0 Delta Delta Delta0 in the Selberg product (10) 1=i 0 (z) Y p 1 Gamma z np j p j : 11) The escape rate (1) is then given by fl = Gamma log 0 . The other 1=i k factors in (10) affect the non leading, anisotropy dependent eigenvalues of L. In particular (see for ex. ref. [20]) the ratio 1 = 0 yields the correlation exponent for chaotic flows. 5 To simplify the discussion of the non leading eigenvalues, in what follows we shall assume that f(x) is a 1 d repelling map, jf 0 (x)j 1, monotone on two nonoverlapping intervals f(x) f 0 (x) x 2 I 0 = f 1 (x) x 2 ....

[Article contains additional citation context not shown here]

F. Christiansen, H.H. Rugh and G. Paladin, submitted to Phys. Rev. A.

.... by the 6 time n 1 e Gammanfl = R V dxdyffi(y Gamma f (n) x) R V dx ; 5) can be extracted from the eigenvalue spectrum of the transfer operator L(y; x) ffi(y Gamma f(x) 6) or its generalizations appropriate to other averages) Physically this spectrum is the correlation [27] [29], resonances [30] quantum energy [31] 36] spectrum, and so on. The spectrum is given by the zeroes of the Fredholm determinant [6, 7] det(1 Gamma zL) expressed in terms of the traces trL n , i.e. the sums over all periodic points x i of period n: trL n = Z dxdyffi(x Gamma y)L n (y; x) ....

....(13) there is an excess in the every order, and the terms cannot be arranged into shadowing combinations. That was the reason why we in ref. 7] concentrated on cycle expansions of the dynamical i functions rather than Selberg products. However, as pointed out by Christiansen, Paladin and Rugh [29] , the shadowing not only works for the Selberg product expansions, but does so with vengeance the convergence with cycle length is faster than exponential. The difference is mathematically innocuous (for hyperbolic systems the Selberg products are entire functions; the dynamical i functions ....

[Article contains additional citation context not shown here]

F. Christiansen, G. Paladin and H.H. Rugh, Nordita preprint (Feb. 1990), submitted to Phys. Rev. A.

.... expansions have also been applied to the irrational windings set of the critical circle maps [9] to the Hamiltonian period doubling repeller [10] to a Hamiltonian three disk pinball [11] to the three disk quantum scattering resonances [12, 13] and to the extraction of correlation exponents [14] . Feasibility of analysis of experimental strange sets in terms of cycles is discussed in ref. 8] 2 ESCAPE RATES A repeller escape rate is an eminently measurable quantity. The experimental measurement consists in shooting many projectiles into a non confining potential and estimating the ....

....f Gamma N X n=nmin c n Gamma A(zc) N 1 1 Gamma zc (38) We shall justify such exponential tail estimates in sect. 7; this particular estimate works if the leading pole is real. Alternatively, one can work with cycle expansions of Selberg products, as discussed in II, sect. 4 and in ref. [14]. Tail resummations often significantly improve the accuracy of the leading root in the cycle expansion; convergence can be further accelerated by Pad e approximants [23] or other acceleration techniques [24] Note also that the existence of a pole at z = 1=c implies that the cycle expansions ....

[Article contains additional citation context not shown here]

F. Christiansen, G. Paladin and H.H. Rugh, Nordita preprint (Jan 1990), submitted to Phys. Rev. A.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC