F. Lucarini, . Freitas, . Freitas, . Holland, . Kuna et al., Let us consider a continuous-time dynamical system? = G(x, t) on a compact manifold Y ? R d , where x(t) = ?(t, t 0 )x(t 0 ), with (s, s) = 1. The two-time evolution operator ? generates a two-parameter semi-group. In the autonomous case, the evolution operator generates a one-parameter semigroup, because of time translational invariance, so that ?(t, s) = ?(t ? s) ?t ? s. In the non-autonomous case, in other terms, there is an absolute clock. We want to consider forced and dissipative systems such that with probability one initial conditions in the infinite past are attracted at time t towards A(t), a time-dependent family of geometrical sets. In more formal terms, we say a family of objects ? t?R A(t) in the finite-dimensional, complete metric phase space Y, Extremes and Recurrence in Dynamical Systems -Chap. 11 -2016/5/16 -19:35 -page 257 257 atisation of these results has been recently proposed in

, t)) Y (P, Q) is the Hausdorff semi-distance between the P ? Y and Q ? Y. We have that d Y (P, Q) = sup x?P d Y (x, Q

Y. , Q. )-=-d-y-(q, P. , Y. , and Q. =-0-?-p-?-q, These can be obtained as evolution at time t through the Ruelle-Perron-Frobenius operator [337] of the Lebesgue measure supported on B in the infinite past, as from the conditions above. Taking the point of view of the chaotic hypothesis, we assume that when considering sufficiently high-dimensional, chaotic and dissipative systems, at all practical levels -i.e. when one considers macroscopic observables -the corresponding measure µ t (dx) is of the SRB type. This amounts to the fact that we can construct at all times t a meaningful (time-dependent) physics for the system, We have that

, 2,2) plot(Zeta,Csi) xlabel('\zeta') ylabel('\csi')

, 2,3) plot(Zeta,Csi) xlabel('\zeta') ylabel('\sigma')

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