Anosov representations can be regarded as a higher-rank version of convex cocompact Kleinian groups. From this point of view, this series of two talks introduces the notion of self-joinings of hyperbolic manifolds, and the dynamics they possess. Mainly, we discuss how the classical result of Sullivan, relating critical exponent and Hausdorff dimension at infinity, is formulated in these higher-rank circumstances. This is joint work with Yair Minsky and Hee Oh.