Integral representation of local and global diffeomorphisms (Q1870869)

The identification of some global diffeomorphisms between closed simply connected sets in \(\mathbb{R}^n\) is the aim of this paper. The author develops in detail the theory of \(C^1\)-diffeomorphisms, derives an integral representation for the elements of \(C^1\) discussing the exponent mapping \(e^T\), \(T\in L(X)\), \(X= \mathbb{R}^n\) or \(\mathbb{C}^n\), derives main results about the integral representation of local diffeomorphisms and also derives the integral representation theorems for global diffeomorphisms on open and closed star-shapen domains in \(\mathbb{C}^n\) and \(\mathbb{R}^n\) in terms of boundary mappings. The essence of these results is to reduce the dimension of the representation problem by passing from the domain to its boundary. This process is used to completely solve -- by induction -- the problem of representation of the global diffeomorphisms on the closed unit ball in \(\mathbb{R}^n\). In this way, the representation in the case \(n=2\) is explicitly constructed and the case of \(C^m\)-diffeomorphisms, \(m> 1\) is briefly discussed. In the last part of the paper, some applications and some concluding remarks are given.