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By P. Collet (auth.), Eric Goles, Servet Martínez (eds.)

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Sci. 20, 130 (1963). [Li) Libchaber, A. , J. Fluid Mech. 204, 1 (1989). S. Young, Ann. of Math. 122,509 (1985). , Y. Pomeau, Commun. Math. Phys. 74, 189 (1980). , USSR Math. Sb. 23, 233 (1974). , Russ. Math. Surv. 32,55 (1977). , Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press, London 1989. Chaotic Evolutions and Strange Attractors. Cambridge University Press, Cambridge 1989. , P. Sulem, J. Phys. 39,441 (1978). , Bull. Amer. Math. Soc. 73, 747 (1967). , Bull. Amer. Math.

This property is called attractivity. This implies that v:::; I" implies vS(t) :::; I"S(t) where v :::; I" is defined by the following equivalent statements 1. There exists v on X 2 with marginals v and I" such that v{(rJ, rJ') : rJ < rJ'} = 1. 2. , v! ' If p :::; A, then vp :::; v>.. To see that, we construct a measure on X 2 with marginals vp and v>. and concentrated on {(rJ, rJ') : rJ :::; rJ'}. We need a family {Ux }xEZ of random variables which are independent and uniformly distributed on [0,1].

The set {w : w(x,l) < t, for all x > O} has probability zero, as well as the event defined in the same way but with x < o. This means that for almost all w there is a pair of sites x, x + 1 such that there are no arrows connecting them in the interval (0, f). Repeating the same argument, we can say that with probability one there is a sequence of sites Xi, i E 2Z such that there are no arrows connecting Xi and Xi + 1 in the time interval (0, f). We consider only the w belonging to this set of probability one.

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