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 f̃ 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− ⊂ à such that B− is bounded to the right, the projection of B− to A is dense, B− − (1, 0) ⊂ B− and f̃ (B−) ⊂ B−. Moreover, if p1 is the projection on the first coordinate of Ã, then there exists d > 0 such that, for any z̃ ∈ B−, lim sup n→∞ p1( f̃ n(z̃))− p1(z̃) 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.