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2
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84927479657
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Lectures in Ergodic Theory, Princeton University, Princeton
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(1977)
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Sinai, Y.1
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4
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84927488732
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ibid.
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(1968)
, vol.2
, pp. 245
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7
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84927481322
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Chaotic Evolution and Strange Attractors, S. Isola, Cambridge University Press, Cambridge
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(1989)
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Ruelle, D.1
Lincee2
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9
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84927497797
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(R) cannot be proved in general. Motion hyperbolicity is usually unknown.
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11
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84927490840
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A task possible in the N=1 case as is considered in [c8] but essentially beyond our capabilities in slightly more general systems, certainly in the nonlinear regime.
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17
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84927495385
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Modeled by a strong wall repulsion.
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19
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84927502046
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Positivity rests on numerical evidence [c12].
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20
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84927469555
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This is usually also assumed in dealing with the ergodic hypothesis, or more generally with principles which cannot be assumed to be independent from the basic laws of motion.
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21
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84927462806
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The assumptions (C) may just be too strong and /or difficult to verify for the model in Eq. 1. Their strength can be seen from the fact that they imply (R) above [c2 c4], and (hence) the ergodic hypothesis if γ=0 [c2 c19]. The density in C could be replaced by the requirement of density in A alone; this would be more general without affecting our conclusions. Furthermore, our main conclusions could still be reached under far weaker assumptions, as we think that the consequences of (C) relevant for our analysis naturally follow in the frame of the theory of singular hyperbolic systems of [c20].
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25
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84927474197
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Lectures at the school “Numerical Methods and Applications,” Granada, Spain, September 1994 [mp_arc@math.utexas.edu, #94-333].
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Gallavotti, G.1
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26
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84927460665
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We identify this with the actual time interval between them, neglecting its fluctuations (which could be easily taken into account leading to the same end result).
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27
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84927487868
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This is just as in the Ising model where one cannot compute correctly the thermodynamic limit average of the magnetization (or the average of any extensive quantity) for a finite volume τ be using the finite volume Gibbs distribution with the same volume τ (without incurring large, size dependent, errors). One must use a larger volume θ≫τ, except in one dimension where it is well known [c4] that the probability distribution of the magnetization would be off “only” by a factor bounded above and below for each spin configuration by a τ-independent constant. Our case is very similar, as follows from the thermodynamic formalism [c2 c4] (see [c22] for a treatment in the frame of the present problem).
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28
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84927481306
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Recall, Ref. [c26], that an observable Gτ has a multifractal behavior if z(β)=limτ→∞ τ-1 ln∫μ¯(dx)Gτ(x)β is not linear in β>0. If the probability for lnGτ(x)∈[p,p+dp]τ is denoted e-ζ(p)τdp, then z(β)=-minp[ζ(p)-βp] and a trivial case arises when ζ(p) has a sharp maximum at some p0, so that z(β)=p0β+const. In our case one experimentally sees the tail of ζ(p); hence our μ¯ is, not surprisingly, a multifractal (i.e., it has a continuum of time scales).
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