Poisson Limit Distribution for Diffeomorphisms with Weak Hyperbolic Product Structure

We study a diffeomorphism which admits a weak hyperbolic product structure region, which is the intersection of two transversal families of weak stable and weak unstable disks, with countably many branches and integrable return times. We show that for such maps the distributions of the number of visits to a ball $B(x, r)$ converges to a Poisson distributions as the radius $r \to 0$. Applications of our resutls are some partially hyperbolic diffeomorphisms of which restriction on one dimensional unstable direction behaves as Manneville-Pomeau maps.