On indecomposable -homogeneous circle-like continua

We exhibit an indecomposable -homogeneous circle-like continuum, answering a question of Neumann-Lara, Pellicer-Covarrubias and Puga-Espinosa from 2005. Our solenoidal construction is an extension of Jim Rogersʼ proof of homogeneity of solenoids of pseudoarcs and leads in turn to an uncountable family of mutually nonequivalent examples. In addition we prove that for an indecomposable -homogeneous continuum Y the two orbits of Homeo(Y) are uncountable.