Homeomorphisms of the annulus with a transitive lift

Let f be a homeomorphism of the closed annulus A that preserves the orientation, the boundary components and that has a lift ${\tilde{f}}$ to the infinite strip à which is transitive. We show that, if the rotation numbers of both boundary components of A are strictly positive, then there exists a closed nonempty unbounded set ${B^{-} \subset \tilde{A}}$ such that B ^? is bounded to the right, the projection of B ^? to A is dense, ${B^{-}-(1, 0) \subset B^{-}}$ and ${\tilde{f}(B^{-}) \subset B^{-}}$ . Moreover, if p ₁ is the projection on the first coordinate of Ã, then there exists d > 0 such that, for any ${\tilde z \in B^{-}}$ , $$\limsup_{n\to\infty}\frac{p_1(\tilde f^n(\tilde z))-p_1(\tilde z)}{n}<-d.$$ In particular, using a result of Franks, we show that the rotation set of any homeomorphism of the annulus that preserves orientation, boundary components, which has a transitive lift without fixed points in the boundary is an interval with 0 in its interior.