An extended definition of Anosov representation for relatively hyperbolic groups

We define a new family of discrete representations of relatively hyperbolic groups which unifies many existing definitions and examples of geometrically finite behavior in higher rank. The definition includes the relative Anosov representations defined by Kapovich-Leeb and Zhu, and Zhu-Zimmer, as well as holonomy representations of various different types of "geometrically finite" convex projective manifolds. We prove that these representations are all stable under deformations whose restriction to the peripheral subgroups satisfies a dynamical condition, in particular allowing for deformations which do not preserve the conjugacy class of the peripheral subgroups.