Quasi-periodic kicking of circle diffeomorphisms having unique fixed

We investigate the dynamics of certain homeomorphisms $F\colon\mathbb{T}^2\to\mathbb{T}^2$ of the form $ F(x,y)=(x+\omega,h(x)+f(y)), $ where $\omega\in\mathbb{R}\setminus \mathbb{Q}$, $f\colon \mathbb{T}\to\mathbb{T}$ is a circle diffeomorphism with a unique (and thus neutral) fixed point and $h\colon \mathbb{T}\to\mathbb{T}$ is a function which is zero outside a small interval. We show that such a map can display a non-uniformly hyperbolic behavior: (small) negative fibred Lyapunov exponents for a.e. $(x,y)$ and an attracting non-continuous invariant graph. We apply this result to (projective) $\mathrm{SL}(2,\mathbb{R})$-cocycles $G\colon (x,u)\mapsto (x+\omega,A(x)u)$ with $A(x)=R_{\phi(x)}B$, where $R_\theta$ is a rotation matrix and $B$ is a parabolic matrix, to get examples of non-uniformly hyperbolic cocycles (homotopic to the identity) with perturbatively small Lyapunov exponents.