text: We consider the general question of estimating decay of correlations for non-uniformly expanding maps, for classes of observables which are much larger than the usual class of Holder continuous functions. Our results give new estimates for many non-uniformly expanding systems, including Manneville-Pomeau maps, many one-dimensional systems with critical points, and Viana maps. In many situations, we also obtain a Central Limit Theorem for a much larger class of observables than usual. Our main tool is an extension of the coupling method introduced by L.-S. Young for estimating rates of mixing on certain non-uniformly expanding tower maps.
  @cite_15: Resume On etudie la vitesse de convergence vers l'etat d'equilibre pour des dynamiques markoviennes non holderiennes. On obtient une estimation de la vitesse de melange sur un sous-espace B dense dans l'espace des fonctions continues. En outre, on montre que le spectre de l'operateur de Perron-Frobenius, restreint aB, est un disque ferme dont chaque point est une valeur propre. Ceci implique que la vitesse de convergence vers l'etat d'equilibre ne peut pas etre exponentielle.
  @cite_14: 
  @cite_4: We present an upper bound on the mixing rate of the equilibrium state of a dynamical system defined by the one-sided shift and a non Holder potential of summable variations. The bound follows from an estimation of the relaxation speed of chains with complete connections with summable decay, which is obtained via a explicit coupling between pairs of chains with different histories.
summary: The have been various results concerned primarily with weakening the assumption on the regularity of @math , and obtaining (slower) upper bounds for the rate of mixing with respect to the corresponding equilibrium measures. Kondah, Maume and Schmitt ( @cite_15 ) used a method of Birkhoff cones and projective metrics, Bressaud, Fernandez and Galves ( @cite_4 ) used a coupling method (different from the one described here), with estimates given in terms of , and Pollicott ( @cite_14 ) introduced a method involving composing transfer operators with conditional expectations. Each of these results has slightly different assumptions and gives slightly different estimates, but in each case a number of different classes of potentials are considered, and estimates are given for for observables of some similar regularity to (usually than) the potential. In particular, in all three examples polynomial mixing is given for a potential and observables with variations decaying at suitable polynomial rates.