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The statistics of attractors

The observation of many experimental systems shows that, even when the time-evolution is described by some deterministic process (e.g. a smooth transformation or an ordinary differential equation), the behaviour of the system may be very hard to understand in deterministic terms, and a stochastic analysis may be a more fruitful approach. Here we restrict ourselves to discrete-time dynamical systems, namely, smooth or piecewise smooth transformations f : M → M on some compact manifold M (possibly with boundary). However, most of the questions and results have natural extensions for flows or semi-flows. A main goal is to describe the evolution of “observable” quantities of the system, that is, of real (or complex) functions φ defined on the phase space M and having some degree of regularity. Quite often, the sequence φ(f (x)) of observations for a typical trajectory f (x) behaves rather erratically as time j varies. This can be illustrated by the following table where the values of φ(f (x)), corresponding to f : [0, 1] → [0, 1], f(x) = 3 ∗ x mod 1, and φ(z) = z, are listed for x = 1/ √ 2 and 0 ≤ j < 60