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 $$f(x)=-\frac{1}{x^a}+\frac{1}{(1-x)^a}$$ , 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 $$A,B\subset [0,1)\times \mathbb {R}$$ , there exists an infinite sequence $$(q_n)_{n=1}^{\infty }$$ such that $$T^{q_n}_f(A)\cap B\ne \emptyset $$ .