On 2009/11/27 at IHP on the occasion of the Katok 65 conference, Patrick BERNARD presented his notion of codimension in Banach spaces with applications to Mather measures and the transversality theorem.
B will denote the ambient Banach space. A large part of the theory extends to Fréchet spaces (things get tougher once C^1 smoothness is involved).
A Lipschitz graph in of codimension is a subset of the graph of a Lipschitz function where and . A rectifiable set of codimension is the image by the canonical projection of a Lipschitz graph of codimension in . A subset of is of codimension if it is a countable union of rectifiable subsets of that codimension.
Theorem (Zajicek 2008). The complement of a subset of positive codimension in a Banach space is Baire generic (i.e., a dense ) and prevalent.
Codimension behaves as expected with respect to C^1 maps: it is invariant under C^1 diffeomorphisms, it decreases the Fredholm index of a C^1 map, it is preserved by preimage under submersions.
It is then strikingly straightforward to prove the following applications:
Theorem. The set of potentials with k+1 Mather measures has codimension k in C^p
Theorem. The set of maps from a m-dimensional manifold to that fail to be injective immersions has codimension n-2k