Lyapunov exponents and rates of mixing for one-dimensional maps. (English) Zbl 1049.37025
Summary: 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.
MSC:
37E05 | Dynamical systems involving maps of the interval |
37A25 | Ergodicity, mixing, rates of mixing |
37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
37E10 | Dynamical systems involving maps of the circle |
37H15 | Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents |