We study attracting graphs of step skew products from the topological andergodic points of view where the usual contracting-like assumptions of thefiber dynamics are replaced by weaker merely topological conditions. In thiscontext, we prove the existence of an attracting invariant graph and study itstopological properties. We prove the existence of globally attracting measuresand we show that (in some specific cases) the rate of convergence to thesemeasures is exponential.
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