# Homeomorphisms of the annulus with a transitive lift - Mathematics > Dynamical Systems

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Abstract: Let $f$ be a homeomorphism of the closed annulus $A$ that preservesorientation, boundary components and that has a lift $\tilde f$ to the infinitestrip $\tilde A$ which is transitive. We show that, if the rotation number ofboth boundary components of $A$ is strictly positive, then there exists aclosed nonempty connected set $\Gamma\subset\tilde A$ such that$\Gamma\subset-\infty,0\times0,1$, $\Gamma$ is unlimited, the projection of$\Gamma$ to $A$ is dense, $\Gamma-1,0\subset\Gamma$ and$\tilde{f}\Gamma\subset \Gamma.$ Also, if $p 1$ is the projection in thefirst coordinate in $\tilde A$, then there exists $d>0$ such that, for any$\tilde z\in\Gamma,$ $$\limsup {n\to\infty}\frac{p 1\tilde f^n\tildez-p 1\tilde z}{n}<-d.$$In particular, using a result of Franks, we show that the rotation set of anyhomeomorphism of the annulus that preserves orientation, boundary components,which has a transitive lift without fixed points in the boundary is an intervalwith 0 in its interior.