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For times much longer than a typical “turnover” time, it makes sense to relax the notion of exact periodicity, and replace it by the notion of recurrence. A point is recurrent or nonwandering if for any open neighborhood M0 of x and any time tmin there exists a later time t, such that f t (x) ∈ M0 . 2) In other words, the trajectory of a non-wandering point reenters the neighborhood M0 inﬁnitely often. We shall denote by Ω the non–wandering set of f , that is the union of all the non-wandering points of M.

4 All equilibrium points are ﬁxed points. Show that a point of a vector ﬁeld v where the velocity is zero is a ﬁxed point of the dynamics f t . 5 Gradient systems.

The formalism should work for any average over any chaotic set which satisﬁes two conditions: 1. the weight associated with the observable under consideration is multiplicative along the trajectory, 2. the set is organized in such a way that the nearby points in the symbolic dynamics have nearby weights. 0, June 18 2003 References 27 transport coeﬃcients and quantum eigenvalues. One of the surprises is that the quantum mechanics of classically chaotic systems is very much like the classical mechanics of chaotic systems; both are described by nearly the same zeta functions and cycle expansions, with the same dependence on the topology of the classical ﬂow.