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The number of periodic points increases exponentially with the cycle length (in the case at hand, as 2n ). As we have already seen, this exponential proliferation of cycles is not as dangerous as it might seem; as a matter of fact, all our computations will be carried out in the n → ∞ limit. Though a quick look at chaotic dynamics might reveal it to be complex beyond belief, it is still generated by a simple deterministic law, and with some luck and insight, our labeling of possible motions will reflect this simplicity.

Miess, Hamiltonian Dynamical Systems (Adam Hilger, Bristol 1987). 15] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge U. Press, Cambridge 1995). 16] S. Smale, “Differentiable Dynamical Systems”, Bull. Am. Math. Soc. 73, 747 (1967). 17] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes in Math. 470 (1975). G. Sinai, “Gibbs measures in ergodic theory”, Russ. Math. Surveys 166, 21 (1972). G. Sinai, “Construction of Markov partitions”, Funkts.

If there exists a connected phase space volume that maps into itself under forward evolution (and you can prove that by the method of Lyapunov functionals, or several other methods available in the literature), the flow is globally contracting onto a subset of M which we shall refer to as the attractor. The attractor may be unique, or there can coexist any number of distinct attracting sets, each with its own basin of attraction, the set of all points that fall into the attractor under foward evolution.

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