Dressed symbolic dynamics

A strange attractor (SA) with symmetry group G can be mapped down to an image strange attractor SA without symmetry by a smooth mapping with singularities. The image SA can be lifted to many distinct structurally stable strange attractors, each equivariant under G, all with the same image SA. If the symbolic dynamics of the image SA requires s symbols σ₁,σ₂,…,σ_s, then |G|s symbols are required for symbolic dynamics in the covers, and there are |G|^s distinct equivariant covers. The covers are distinguished by an index. The index is an assignment of a group operator to each symbol σ_i:σ_i→g_αi. The subgroup H⊂G generated by the group operators g_αi in the index determines how many disconnected components (|G|/|H|) each equivariant cover has. The components are labeled by coset representatives from G/H. The structure of each connected component is determined by H. A simple algorithm is presented for determining the number and the period of orbits in an equivariant attractor that cover an orbit of period p in the image attractor. Modifications of these results for structurally unstable covers are summarized by an adjacency diagram.