# Borel equivalence relations by Vladimir Kanovei

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56) It is not the case for a mixing trajectory when N (t) grows faster than t. This new feature of the mixing dynamics can be characterized by the dynamical entropy if the number of cells grows exponentially with time. Fig. 4. Two cases of coarse-graining: Stable trajectory (left) of nonergodic dynamics and unstable trajectory (right) with mixing. 2 Kolmogorov–Sinai (KS) entropy Let us ﬁnd a link between partitioning and time. 57) which is discrete for a simplicity n ∈ IN . Let us ﬁx a time n and partition (n) (n) ξ (n) = (ξ1 , ξ2 , .

The described set of islands will be called resonant islands or chain of islands. Their study leaves aside many questions, and particularly the question of a role of all other possible resonances. 4). 4 describe how the perturbation theory works, at least in a simpliﬁed way. The role of neglected terms can be estimated for a ﬁnite time. 5). The crucial idea of the KAM theory is to consider invariant tori rather than trajectories and to study a perturbation of the tori rather than stability of trajectories.

A more frequently used characteristic is Lyapunov exponent. 17). The Jacobian matrix M ˆ = DTˆxn = M ∂xi;n+1 ∂xj;n , (i, j = 1, . . 24) where N is the number of degrees of freedom. According to the Liouville theorem ˆ | = 1 for any n. |M Consider the expression ˆ n = Dv Tˆn x, M where v denotes a vector along which the derivative is taken. 25) form an ‘ellipsoid’ of values λ with a centre at x and a fan of directions v. e. σk (x) = lim n→∞ 1 ln |Dvk Tˆn x|, n k = 1, . . 26) where vk deﬁnes orthogonal directions of maximal rates of dispersion of trajectories.