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Posts Tagged ‘prevalence’

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 B of codimension k is a subset of the graph of a Lipschitz function C \to T where B=C\oplus T and \dim(T)=k. A rectifiable set of codimension k is the image by the canonical projection B\times \mathbb R^n\to B of a Lipschitz graph of codimension k+n in B\times\mathbb R^n.  A subset of B is of codimension k 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 G_\delta) 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 \mathbb R^n that fail to be injective immersions has codimension n-2k

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