Decay of Correlations

1. What are these lectures about? Invariant measures, physical measures, and rates of mixing We are interested in discrete-time dynamical systems represented by the iterates f n = f f n?1 (where n 2 Z + represents time) of a transformation f : M ! M which is \chaotic," in the sense that arbitrarily close distinct initial points become \separated" (assuming a metric structure or at least a topology on M) if one waits long enough. This separation (sometimes referred to as \sensitive dependence on initial conditions") often takes place at exponential speed, as in the elementary paradigm of the \angle-doubling" map z 7 ! z 2 on the circle fz 2 C j jzj = 1g, where the distance between f n x and f n y is 2 n times the distance between x and y if they are close enough. Our aim is to describe the long-time behaviour of generic initial conditions. Generic is understood in a measure-theoretical sense, so that this task of statistically describing the asymptotics of \most" initial data is not rendered completely hopeless by the sensitive dependence on initial conditions. We shall mostly discuss the case when M is a compact Riemann manifold, so that we have a natural a priori probability measure on M: the Lebesgue measure. We therefore seek to understand invariant probability measures (i.e., (f ?1 (E)) = (E) for every Borel set) which are somehow related to the a priori measure. The rst key concept is, of course, ergodicity (see Wal] for this, and other, ergodic-theoretical notions). Recall that the celebrated Birkhoo theorem says that if is an f invariant Borel measure such that (f;) is ergodic (i.e., \indecomposable" in the sense that E = f ?1 E only if (E) = 0 or 1), then for each continuous \test" function (also called \observable") ' : M ! C and-almost all x 2 M the \time average converges to the space average:" lim n!1 1 n n?1 X i=0 '(f i x) = Z ' dd : (1.1) (In fact, convergence does not hold only for continuous observables, L 1 (dd) would suuce.) Of course, can be supported on a set of zero Lebesgue measure (the