A Classification of Toral and Planar Attractors and Substitution Tiling Spaces

We focus on dynamical systems which are one-dimensional expanding attractors with a local product structure of an arc times a Cantor set. We define a class of Denjoy continua and show that each one of the class is homeomorphic to an orientable DA attractor with four complementary domains which in turn is homeomorphic to a tiling space consisting of aperiodic substitution tilings. The planar attractors are non-orientable as is the Plykin attractor in the 2-sphere which we describe. We classify these attractors and tiling spaces up to homeomorphism and the symmetries of the underlying spaces up to isomorphism. The criterion for homeomorphism is the irrational slope of the expanding eigenvector of the defining matrix from whence the attractor was formed whilst the criterion for isomorphism is the matrix itself. We find that the permutation groups arising from the 4 'special points' which serve as the repelling set of an attractor are isomorphic to subgroups of S[subscript 4]. Restricted to these 4 special points, we show that the isotopy class group of the self-homeomorphisms of an attractor, and likewise those of a tiling space, is isomorphic to Z ⊕ Z[subscript 2].