On the number of orbits of the homeomorphism group of solenoidal spaces

A continuum X is called solenoidal if it is circle-like and nonplanar. X is 1n-homogeneous if the action of its homeomorphism group Homeo(X) on X has exactly n orbits; i.e. there are exactly n types of points in X. Recently Jiménez-Hernández, Minc and Pellicer-Covarrubias [6] constructed a family of 1n-homogeneous solenoidal continua, for every n>2. Modifying the spaces obtained by them, as well as an earlier construction of the author for n=2, for every n>2 we construct two different uncountable families of arcless 1n-homogeneous solenoidal continua Σn and Σn′. We also show that there is an uncountable family of countably nonhomogeneous solenoidal continua Σ∞; i.e. each Y∈Σ∞ has (infinitely) countably many types of points. For every Y∈⋃n∈NΣn∪Σ∞ any orbit of Homeo(Y) is uncountable. With respect to the degree of homogeneity, in the realm of solenoidal continua containing pseudoarcs, our examples complete the gap between homogeneous solenoids of pseudoarcs and uncountably nonhomogeneous pseudosolenoids. A number of questions related to the study is raised.