We introduce the notion of restricted Anosov representations, characterized by their dominated splitting behavior over associated flows, to encompass many non-Anosov representations with good geometric properties, such as Minsky's primitive-stable representations. As a first application, for a closed hyperbolic surface group, we show that the collection of representations which are Anosov in restriction to the simple geodesics flow gives a domain of discontinuity for the mapping class group action (joint with Nicolas Tholozan). Secondly, for a relatively hyperbolic group, we show that a representation being both divergent and Weisman's extended geometrically finite is equivalent to being Anosov in restriction to a flow associated with the boundary extension.
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