Abstract:
With a view toward the preparation of the apparatus for the investigation of quasiconformal mappings of manifolds, in this work we establish the following local variant of M. A. Lavrent'ev's theorem concerning a global homeomorphism proved earlier by us.
Theorem. {\it Let $F$ be a locally homeomorphic mapping of the deleted sphere $\Dot B=\{x\mid0<|x|<r_0\}\subset\mathbf R^n$ into $\mathbf R^n$. Let $k(r)$ be the coefficient of quasiconformality of $F$ in the region $\{x\mid0<r<|x|<r_0\}$. Then the following assertions are valid.
$1^\circ)$ When $\int_0\frac1{rk(r)}\,dr=\infty$ and $n\geqslant3,$ the mapping $F$ is homeomorphic in some deleted neighborhood of the point $x=0,$ and can be continued up to homeomorphism to the whole neighborhood of this point.
$2^\circ)$ In the sense of the admissible order of the growth of $k(r),$ the assertion $1^\circ)$ is correct}.
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