In this paper we construct a family of circle-like continua, each admitting a finest monotone map onto S1 such that there exists a subset of point inverses which is homeomorphic to the Cantor set cross an interval. We then show how to realize some members of this family as the boundaries ∂U of bounded irreducible local Siegel disks U. These boundaries are geometrically rigid in the following sense: there exist arbitrarily small periodic homeomorphisms of the sphere, conformal on U, which keep Ū invariant. The embedding portion of this paper follows a flexible construction of Herman. These results provide a partial answer to a question of Rogers and a complete answer to a question of Brechner, Guay, and Mayer. © 2001 Elsevier Science B.V. All rights reserved.