Mixing for Some Non-Uniformly Hyperbolic Systems

In this work, we present an abstract framework that allows to obtain mixing (and in some cases sharp mixing) rates for a reasonable large class of invertible systems preserving an infinite measure. The examples explicitly considered are the invertible analogue of both Markov and non-Markov unit interval maps. For these examples, in addition to optimal results on mixing and rates of mixing in the infinite case, we obtain results on the decay of correlation in the finite case of invertible non-Markov maps, which, to our knowledge, were not previously addressed. The proposed method consists of a combination of the framework of operator renewal theory, as introduced in the context of dynamical systems by Sarig (Invent Math 150: 629-653, 2002), with the framework of function spaces of distributions developed in the recent years along the lines of Blank et al. (Nonlinearity 15: 1905-1973, 2001).