Ergodic theory and dynamical systems (original) (raw)

1985, Advances in Mathematics

We show that one-dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive, some power of f is mixing and, in particular, the correlation of Hölder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, by the average rate at which typical points start to exhibit exponential growth of the derivative. http://journals.cambridge.org Downloaded: 07 Dec 2012 IP address: 200.128.60.106 638 J. F. Alves et al 1.2. Decay of correlations. Positive Lyapunov exponents are known to be a cause of sensitive dependence on initial conditions and other dynamical features which give rise to a degree of chaoticity or stochasticity in the dynamics. We can formalize this idea through the notion of mixing with respect to some invariant measure. Definition 2. A probability measure µ defined on the Borel sets of I is said to be f-invariant if µ(f −1 (A)) = µ(A) for every Borel set A ⊂ I. Definition 3. A map f is said to be mixing with respect to some f-invariant probability measure µ if |µ(f −n (A) ∩ B) − µ(A)µ(B)| → 0, when n → ∞, for any measurable sets A, B. One interpretation of this property is that the conditional probability of B given f −n (A), i.e. the probability that the event A is a consequence of the event B having occurred at some time in the past, is asymptotically the same as if the two events were completely independent. This is sometimes referred to as a property of loss of memory, and thus in some sense of stochasticity, of the system. A natural question of interest both for application and for intrinsic reasons, therefore, is the speed at which such loss of memory occurs. Standard counterexamples show that, in general, there is no specific rate: it is always possible to choose sets A and B for which mixing is arbitrarily slow. However, this notion can be generalized in the following way. Definition 4. For a map f : I → I preserving a probability measure µ and functions ϕ, ψ ∈ L 1 (µ), we define the correlation function C n = C n (ϕ, ψ) = (ϕ • f n)ψ dµ − ϕ dµ ψ dµ .