Enveloping manifolds

We study the problem of embedding compact subsets of R n into C 1 submanifolds of minimal dimension. In [Ott and Yorke, SIAM J. Appl. Dynamical Systems 2 (2003) 297], we define a generalized tangent space T x A suitable for a general compact subset A of R n and we prove that A may be locally embedded into a C 1 manifold of dimension dim ( T x A ) . This result leads naturally to the global conjecture that for a compact subset A of R n , there exists a C 1 manifold M such that M ⊃ A and dim M = max x ∈ A dim ( T x A ) . We prove that this conjecture is false in general, but true if dim ( T x A ) is constant on A. Applications of these ideas to dimension theory, embedding theory, and dynamical systems are discussed.