The main goal in this paper is to study attracting invariant multi-graphs for a certain class of skew products. An invariant multi-graph is an invariant compact set which is a finite union of invariant graphs, and thus consists of a finite number of points on each fiber.We introduce invariant bony multi-graphs and construct an open set of skew products over an invertible base map (solenoid map) having attracting invariant multi graphs and bony multi-graphs. These multi-graphs are the support of finitely many ergodic SRB measures. In this study some thermodynamic properties are investigated for these systems.We will provide some sufficient conditions ensuring the existence of equilibrium states supported on invariant multi-graphs. Finally, we extend our results to a family of skew products over a generalized baker map.

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