Homeomorphisms of the annulus with a transitive lift

Abstract

Let f be a homeomorphism of the closed annulus A that preserves the orientation, the boundary components and that has a liftf to the infinite stripà 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 setif p 1 is the projection on the first coordinate ofÃ, then there exists d > 0 such that, for anỹIn 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.

Transitivity and rotation sets with nonempty interior for homeomorphisms of the $2$-torus

“…The work [4] suggests this is also true C 1 generically. In [1] it is shown that, if g is a homeomorphism of the closed annulus whose lift g to the strip R × [0, 1] is transitive, and if there are no fixed points in the boundary of the annulus, then 0 is in the interior of the rotation interval of g. Some of the ideas presented in this paper follow from that paper.…”

“…First, note that the orbit of x T is not bounded in any direction, and as such If tg(θ) is rational, then the result follows from Proposition 5 by taking v ∈ Z 2 perpendicular to e θ , since each of the points x T + iv, i ∈ Z needs to be in a different connected component of O 1 .…”

“…Therefore, there is a point y in π −1 (x) such that y; e θ ≥ (p 2 , q 2 ); e θ + 3. This means that the points y, y + (0, 1) and y − (1, 0) are all in the semi-plane V We construct H in the following way: Let H(0) = y, H| [0,1] (t) = h 1 (t). For i ∈ Z, we define H| [i,i+1] (t) = h 1 (t + i) if H(i); e θ < y; e θ , and H| [i,i+1] …”

“…Given an area-preserving homeomorphism of the torus f and a fixed lift f, one can consider the rotation vector of the Lebesgue measure λ for f, 1] f (x) − xdλ.…”

On generic rotationless diffeomorphisms of the annulus

“…In this subsection we define the set B − , introduced in [1], that will play an important role in the proof of our theorem. Although much of what is done in this subsection can be found in [1], for completeness sake we present all results needed with proofs.…”