Realizing rotation numbers on annular continua

An annular continuum is a compact connected set $K$ which separates a closed annulus $A$ into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the case where $K=A$, showing that if $K$ is an invariant annular continuum of a homeomorphism of $A$ isotopic to the identity, then the rotation set in $K$ is closed. Moreover, every element of the rotation set is realized by an ergodic measure supported in $K$ (and by a periodic orbit if the rotation number is rational) and most elements are realized by a compact invariant set. Our second result shows that if the continuum $K$ is minimal with the property of being annular (what we call a circloid), then every rational number between the extrema of the rotation set in $K$ is realized by a periodic orbit in $K$. As a consequence, the rotation set is a closed interval, and every number in this interval (rational or not) is realized by an orbit (moreover, by an ergodic measure) in $K$. This improves a previous result of Barge and Gillette.