On -homogeneous continua

A continuum is -homogeneous provided there are exactly two orbits for the action of the group of homeomorphisms of the continuum onto itself. In this paper we study some relations between -homogeneous continua and their set of cut points. We also prove that if X is a hereditarily decomposable continuum whose proper, nondegenerate subcontinua are arc-like, then X is -homogeneous if and only if X is an arc. Suitable examples and counterexamples are given.