Let $\mathcal{A}$ be a smooth maximal atlas on a <s>differentiable</s> manifold $M$. Let $f:M\to M$ be a smooth invertible function, whose inverse is not smooth (for example $f:\mathbb R\to \mathbb R$, $f(x)=x^3$). Then $f$ induces an atlas $\mathcal{A}'$ which is not compatible with $\mathcal{A}$.

Since the relation of compatibility between <s>such</s> atlases is an equivalence, the problem of classifying them appears. To simplify, we can try to classify the smooth atlases which are compatible on $M-N$, where $N\subset M$, but are incompatible on $N$. The simplest case seems to be when $N$ contains only a point.

Another problem is to find all maximal atlases for which a given function on $M$, or other object, for example a tensor, is smooth.

Are there any studies of these kinds of classifications of atlases?
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**Update:** I don't ask about exotic smooth structures. The compatibility of atlases is "finer", but both are named "smooth structures", which leads to confusions (see [http://en.wikipedia.org/wiki/Smooth_structure#Confusion_about_terminology][1])


  [1]: http://en.wikipedia.org/wiki/Smooth_structure#Confusion_about_terminology