Representation theory for strange attractors

Embeddings are diffeomorphisms between some unseen physical attractor and a reconstructed image. Different embeddings may or may not be equivalent under isotopy. We regard embeddings as representations of the attractor, review the labels required to distinguish inequivalent representations for an important class of dynamical systems, and discuss the systematic ways inequivalent embeddings become equivalent as the embedding dimension increases until there is finally only one “universal” embedding in a suitable dimension.

Figure 1

Projection of the knot-type isotopy into ${\mathbb{R}}^3$.

Figure 2

Isotopy of two Dehn twists to the identity in ${\mathbb{R}}^4$.