Cantor sets of arcs in decomposable local Siegel disk boundaries☆☆Portions of this paper were presented by the first author at the SE Sectional Meeting of the AMS (Gainesville, FL, March 1999), and at the Spring Topology and Dynamics Conference (Salt Lake City, UT, March 1999).

AbstractIn 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 U 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