encoding dynamics

Given a "data manifold" M⊂ ℝ^n and "latent space" ℝ^ℓ, an autoencoder is a pair of continuous maps consisting of an "encoder" E: ℝ^n→ ℝ^ℓ and "decoder" D: ℝ^ℓ→ ℝ^n such that the "round trip" map D E is as close as possible to the identity map id_M on M. We present various topological limitations and capabilites inherent to the search for an autoencoder, and describe capabilities for autoencoding dynamical systems having M as an invariant manifold.