This paper investigates the geometric structures of non-uniformly hyperbolic attractors of a certain class of skew products. We construct an open set of skew products with the fiber an interval over a linear expanding circle map such that any skew product belonging to this set admits a non-uniformly hyperbolic solenoidal attractor for which the following dichotomy is ascertained. This attractor is either the image of a continuous invariant graph under a semi conjugacy with nonempty interior, the so-called massive attractor, or a thick generalized bony attractor. Here, a generalized bony attractor is, roughly speaking, the image of a bony graph attractor under a semi conjugacy. Also, an attractor is thick if it has positive but not full Lebesgue measure. Moreover, in both cases, the attractors are mixing. In our construction, the contraction in the fiber is non-uniform and hence it is specified in terms of fiber Lyapunov exponents. Furthermore, we provide some related results on the ergodic properties of attracting graphs and stability results for such graphs under deterministic perturbations. In particular, we show that there exists an invariant ergodic SRB measure whose support is contained in that attractor.

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