Let A be an annulus in the plane \({\mathbb {R}}^2\) and \(g:A\rightarrow A\) be a boundary components preserving homeomorphism which is distal and has no periodic points. Then there is a continuous decomposition of A into g-invariant circles such that all the restrictions of g on them share a common irrational rotation number and all these circles are linearly ordered by the inclusion relation on the sets of bounded components of their complements in \({\mathbb {R}}^2\) . Finally, we show that g is conjugate to an irrational rotation if and only if there exists a transversal and examples without transversals are also constructed.
