Dressed symbolic dynamics

A strange attractor (SA) with symmetry group $\mathcal{G}$ 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 $\mathcal{G},$ all with the same image SA. If the symbolic dynamics of the image SA requires s symbols ${\sigma }_1,{\sigma }_2,\dots ,{\sigma }_s,$ then $|\mathcal{G}|s$ symbols are required for symbolic dynamics in the covers, and there are $|\mathcal{G}{|}^s$ distinct equivariant covers. The covers are distinguished by an index. The index is an assignment of a group operator to each symbol ${\sigma }_i:{\sigma }_i\to g_{{\alpha }_i}.$ The subgroup $\mathcal{H}\subset G$ generated by the group operators $g_{{\alpha }_i}$ in the index determines how many disconnected components $(|\mathcal{G}|/|\mathcal{H}|)$ each equivariant cover has. The components are labeled by coset representatives from $\mathcal{G}/\mathcal{H}.$ The structure of each connected component is determined by $\mathcal{H}.$ 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.