Let $\mathcal{A}$ be a smooth maximal atlas on a differentiable 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 such 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?

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).

Update 2 (example): Let's consider the manifold $\mathbb R$ with the maximal atlas $\mathcal A$ generated by $id:\mathbb R\to \mathbb R$. Let the atlas $\mathcal A'$ be generated by $f(x)=x^3$. Then not all functions on $\mathbb R$ which are smooth in one atlas are smooth in the other too.

Now let's consider the set $N$ containing only the origin. When we restrict the charts from $\mathcal A$ and $\mathcal A'$ to $M-N$, we obtain two compatible atlases, although $\mathcal A$ and $\mathcal A'$ are not compatible. This justifies the first problem:

Q1: Find all the maximal atlases on $\mathbb R$ whose restriction on $M-N$ is compatible with $\mathcal A|_{M-N}$.

Since not all functions on $\mathbb R$ which are smooth in one atlas are smooth in the other, the following problem arises:

Q2: Find all the maximal atlases in which a given function is smooth.