Covering dynamical systems: Twofold covers

We study the relation between a dynamical system, which is unchanged (equivariant) under a discrete symmetry group G and another locally identical dynamical system with no residual symmetry. We also study the converse mapping: lifting a dynamical system without symmetry to a multiple cover, which is equivariant under G. This is done in $R^3$ for the two element rotation and inversion groups. Comparisons are done for the equations of motion, the strange attractors that they generate, and the branched manifolds that classify these strange attractors. A dynamical system can have many inequivalent multiple covers, all equivariant under the same symmetry group G. These are distinguished by the value of a certain topological index. Many examples are presented. A new global bifurcation, the “peeling bifurcation,” is described.