Consider a homeomorphism of closed surface of genus $g \ge 2$. I will explain that in the case its homological rotation set (a compact subset of $\mathbf R^{2g}$ capturing the rotational behaviour of the dynamics) is big enough, the whole rotational behaviour is contained in a compact set that resembles a finite union of homoclinic classes with some heteroclinic connections. This is related to $C^0$ rotational versions of properties like Markov partitions, rotational density of periodic orbits, stability under perturbations… as well as purely rotational features such as the description of the rotation set’s shape or some bounded deviations properties. The whole thing is based on Le Calvez-Tal forcing theory but I will mainly focus on some examples.