Mixing properties of some maps with countable Markov partitions

In the previous works of the author and S. Newhouse [Trans. Amer. Math. Soc. 171 (1996), pp. 89-105] and [Asterisque, 261 (2000), pp. 103-16] a class of piecewise smooth two-dimensional systems with countable Markov partitions was studied, and Bernoulli property was proved. In this paper we consider 2-d maps F satisfying the same hyperbolicity and distortion conditions, and assume similar conditions for F-1. We assume additionally that contraction of each map increases when points approach the boundary of its domain. For such systems we extend the results of the author [Contemp. Math. 692 (2017), pp. 177-193], and prove exponential decay of correlations.