Many problems in dynamical systems deal with the study of the limit behavior of orbits, leading to the study of the so-called attractors. Understanding the structure of attractors is a major goal in the area. We consider random dynamical systems forced by a deterministic external factor. In general, systems of this kind are modeled, in discrete time, as skew products. Skew products are used as a tool for the construction of robust or generic examples of complicated behaviors. Here, we study invariant graphs which are the natural substitutes of attracting fixed points of autonomous systems and they play a central role in the theory. The main goal is to obtain skew products having attracting invariant graphs with a complicated dynamics exhibiting nice ergodic properties such as the existence of SRB measures whose supports lie on invariant graphs. The other goal here is to describe the structure of invariant graphs and study the properties of the pinching set, the set of points where the values of all of the invariant graphs coincide.

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