Equilibrium states with polynomial decay of correlations

Abstract. The purpose of this paper is to study statistical properties of some al-most expanding dynamical systems. Examples of such systems include piecewise expanding maps on the unit interval, expanding maps on Cantor sets, and some piecewise expanding maps on the unit cubes, all of which contain an indierent xed point. We suppose the systems have Markov partition. So we can work on symbolic dynamical systems. We prove existence of absolutely continuous invariant measures with respect to either Lebesgue measures or conformal measures, and show that these measures are weak Gibbs states and equilibrium states. Further, by using the projective metric we obtain that under iteration of the Perron-Frobenius operators, functions converge to the equilibriums in polynomial speed, and therefore the systems have polynomial decay of correlations. The motivation of the paper is to understand statistical properties for almost ex-panding dynamical systems that allow absolutely continuous probability invariant measures with respect to Lebesgue or conformal measures. We say that a piecewis