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We prove a structure theorem for ergodic homological rotation sets ofhomeomorphisms isotopic to the identity on a closed orientable hyperbolicsurface: this set is made of a finite number of pieces that are eitherone-dimensional or almost convex. The latter ones give birth to horseshoes; inthe case of a zero-entropy homeomorphism we show that there exists a geodesiclamination containing the directions in which generic orbits with respect toergodic invariant probabilities turn around the surface under iterations of thehomeomorphism. The proof is based on the idea of $\textit{geodesic tracking}$of orbits that are typical for some invariant measure by geodesics on thesurface, that allows to get links between the dynamics of such points and theone of the geodesic flow on some invariant subset of the unit tangent bundle ofthe surface.

Geodesic tracking and the shape of ergodic rotation sets