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Inspired by a twist maps theorem of Mather we study recurrent invariant setsthat are ordered like rigid rotation under the action of the lift of a bimodalcircle map $g$ to the $k$-fold cover. For each irrational in the interior ofthe rotation set the collection of the $k$-fold ordered semi-Denjoy minimalsets with that rotation number contains a $(k-1)$-dimensional ball in the weaktopology on their unique invariant measures. We also describe completely theirperiodic orbit analogs for rational rotation numbers. The main tool is ageneralization of a construction of Hedlund and Morse which generates thesymbolic analogs of these $k$-fold well-ordered invariant sets.

On the abundance of $k$-fold semi-monotone minimal sets in bimodal circle maps