One can study a dynamical system taking several points of view. On the one hand one can observe behavior of distinguished orbits and, for example, derive existence of periodic orbits, their distribution in phase space as well as their in-/stability properties (topological viewpoint of dynamics). On the other hand one can investigate "probabilistic" behavior and study probability measures which are invariant under the dynamics and their properties (viewpoint from ergodic theory) and again study the space of all such measures. In the context of differentiable uniformly hyperbolic maps (such as the horseshoe map) it is well known that periodic orbits are dense in phase space and that on the other hand the corresponding invariant (ergodic) measures are dense in the space of all invariant measures. One can now prove - under relatively general conditions - that it holds a kind of reverse relation between these objects: Suppose that a convex sum of two ergodic hyperbolic measures supported on a periodic hyperbolic orbit is arbitrarily well approximated by ergodic measures then both periodic orbits belong to a common horseshoe. (Joint work with Ch. Bonatti)