On the rotation sets of generic homeomorphisms on the torus

We study the rotation sets for homeomorphisms homotopic to the identity on the torus Tᵈ , d≥ 2 . In the conservative setting, we prove that there exists a Baire residual subset of the set Homeo0, λ( T²) of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in T² , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every d≥ 2 the rotation set of C⁰ -generic conservative homeomorphisms on Tᵈ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.