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 \emph{topologically recurrent}, i.e. for every open rectangle $A\subset [0,1)\times \mathbb{R}$, there exists an infinite sequence $(q_n{)}_{n=1}^{\mathrm{\infty }}$ such that $T_f^{q_n}(A)\cap A\mathrm{\ne }\mathrm{\varnothing }$.