Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity

We show that if a homeomorphism f of the torus, isotopic to the identity, has three or more periodic orbits with non-collinear rotation vectors, then it has positive topological entropy. Furthermore, for each point ρ of the convex hull Δ of their rotation vectors, there is an orbit of rotation vector ρ, for each rational point p / q , p ∈ ℤ 2 , q ∈ ℕ, in the interior of Δ, there is a periodic orbit of rotation vector p / q , and for every compact connected subset C of Δ there is an orbit whose rotation set is C . Finally, we prove that f has ‘toroidal chaos’.