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The inverse function theorem states that:

Suppose that $f: U \subset \mathbb{R}^m \rightarrow \mathbb{R}^m$ is a $C^k$-function and that there exists $a \in U$ such that $f'(a): \mathbb{R}^m \rightarrow \mathbb{R}^m $ is an isomorphism. Then, there exist $\delta > 0$ and an open ball $B_{\delta} : = B(a, \delta) \subset U$ such that $f \mid_{B_{\delta}} : B_{\delta} \rightarrow V \ni f(a)$ is a diffeomorphism, with $V$ being an open set.

There are two remarkable applications of this theorem:

1- Existence of matrices $X$ such that $X^ k = Y$ where $Y$ is a matrix sufficiently close to the identity;

2- Differentiable perturbation of the identity: Let $U \subset \mathbb{R}^m$ a convex and open set. If $ \varphi : U \rightarrow \mathbb{R}^m$ is $C^k$, with $|\varphi'(x)| \leq \lambda < 1$ for all $x \in U$, then $f : U \rightarrow \mathbb{R}^m$ given by $f(x) = x + \varphi(x)$ is a diffeomorphism on $U$ onto its image $f(U)$.

My goal with this question is to broaden the knowledge about the application/importance of this theorem in other contexts in the areas of Analysis, Geometry, Differential Topology, etc...

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  • $\begingroup$ This is a subjective question and has no concrete, acceptable answer … $\endgroup$ – k.stm Nov 13 '19 at 23:21
  • $\begingroup$ Dear @k.stm, from my view point your claim no souds good. In fact if someone give new/ interesting examples of application of this theorem will be a gain for community, such examples will be in fact concrete answers. $\endgroup$ – Marcelo Ng Nov 14 '19 at 15:09
  • $\begingroup$ hint: dig in math.stackexchange.com/questions/tagged/… $\endgroup$ – janmarqz Dec 1 '19 at 0:44

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