Dendroids with a low homogeneity degree

The homogeneity degree of a topological space X is the number of orbits of the action of the homeomorphism group of X on X. The property of low homogeneity degree offers an analog of homogeneity within classes of spaces which do not contain any homogeneous spaces, such as dendroids. In this paper, we study dendroids of low homogeneity degree. We show that the arc is the only dendroid with homogeneity degree 2, and prove several partial results towards a classification of smooth dendroids of homogeneity degree 3.