Representation theory of dynamical systems

In physics, experiments form the bridge connecting theory to reality. This bridge is often quite narrow: one typically only records one of the myriad variables responsible for generating a complicated dynamical behavior. Nevertheless, every variable can usually be "reconstructed" from that single observation. Such a reconstruction provides an embedding of the original phase space into some Euclidean space. However, this reconstruction or embedding is not unique. Most analyses of experimental data from complex dynamical systems depend on these reconstructions, and they could, in principle, depend on the choice of reconstruction. It is the purpose of this thesis to establish a framework suitable to address this dependence on reconstruction: a representation theory for dynamical systems. This dynamical representation theory is constructed in analogy with the well known representation theory for groups. We regard reconstructions or embeddings as representations of dynamical phase spaces. Different embeddings may or may not be equivalent under smooth deformations. The program of representation theory is to work out all inequivalent representations in Euclidean spaces of various dimension and to identify the topological obstructions preventing equivalence between distinct representations. Our main result is the complete representation theory for three dimensional dynamical systems. This is possible because the theory of three dimensional systems is rather well understood. In contrast, higher dimensional systems are much less thoroughly understood, so we offer only preliminary results for the representation theory of a certain class of dynamical systems that exist in every dimension. We also present some results for equivariant dynamical systems. While not a part of representation theory proper, these investigationswere motivated by a problem that manifests only when viewing reconstructions of the Lorenz system from the perspective of representation theory.