Coexistence of two contrasting recurrence properties of certain non-integrable cocycles

We study the recurrence properties of certain skew products over symmetric interval exchange transformations, including rotations, with cocycles of the form f(x)=-1/xᵃ+1/(1-x)ᵃ , where $$a>1$$ . We prove that typically, such systems are dissipative. However, at the same time they are topologically transitive, i.e. for every two open rectangles A,B⊂ [0,1)× R , there exists an infinite sequence (qₙ)ₙ₌₁∞ such that Tqₙf(A)∩ B≠ ∅ .