Anosov representations, strongly convex cocompact groups and weak eigenvalue gaps

We provide characterizations of Anosov representations of word hyperbolic groups into real semisimple Lie groups in terms of equivariant limit maps, the Cartan property and the uniform gap summation property of \cite{GGKW}. As an application we obtain a characterization of strongly convex cocompact subgroups of the projective linear group $\mathsf{PGL}(d, \mathbb{R})$. We also compute the Holder exponent of the Anosov limit maps of an Anosov representation in terms of the Cartan and Lyapunov projection of the image.