Let $\mathcal{A}$ be a smooth maximal atlas on a 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 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.