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Difference between revisions of "Zorich theorem"

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In 1967, V.A. Zorich proved the following result for quasi-regular mappings in space (see also [[Quasi-regular mapping|Quasi-regular mapping]]): A locally homeomorphic quasi-regular mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301402.png" />, is, in fact, a [[Homeomorphism|homeomorphism]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301403.png" />.
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In 1967, V.A. Zorich proved the following result for [[quasi-regular mapping]]s in space: A [[Local homeomorphism|locally homeomorphic]] quasi-regular mapping $f : \mathbf{R}^n \rightarrow \mathbf{R}^n$, $n \ge 3$, is, in fact, a [[homeomorphism]] of $\mathbf{R}^n$.
  
This result had been conjectured by M.A. Lavrent'ev in 1938. Note that the exponential function shows that there is no such result for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301404.png" />. In 1971, O. Martio, S. Rickman and J. Väisälä proved a stronger quantitative result: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301406.png" /> there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301407.png" />, the radius of injectivity, such that every locally injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301408.png" />-quasi-regular mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z13014010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z13014011.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z13014012.png" />, is injective in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z13014013.png" />.
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This result had been conjectured by M.A. Lavrent'ev in 1938. Note that the exponential function shows that there is no such result for $n=2$. In 1971, O. Martio, S. Rickman and J. Väisälä proved a stronger quantitative result: For $n \ge 3$ and $K > 1$ there exists a number $\psi(n,K) \in (0,1)$ , the ''radius of injectivity'', such that every locally injective $K$-quasi-regular mapping $f : B^n \rightarrow \mathbf{R}^n$, where $B^b = B^n(1)$ and $B^n(r) = \{ x \in \mathbf{R}^n : |x| \le r \}$, for $r > 0$, is injective in $B^n(\psi(n,K))$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Rickman,  "Quasiregular mappings" , ''Ergeb. Math. Grenzgeb.'' , '''26''' , Springer  (1993)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.A. Zorich,  "The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems"  M. Vuorinen (ed.) , ''Quasiconformal Space Mappings'' , ''Lecture Notes in Mathematics'' , '''1508'''  (1992)  pp. 132–148</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Martio,  U. Sebro,  "Universal radius of injectivity for locally quasiconformal mappings"  ''Israel J. Math.'' , '''29'''  (1978)  pp. 17–23</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Rickman,  "Quasiregular mappings" , ''Ergeb. Math. Grenzgeb.'' , '''26''' , Springer  (1993)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  V.A. Zorich,  "The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems"  M. Vuorinen (ed.) , ''Quasiconformal Space Mappings'' , ''Lecture Notes in Mathematics'' , '''1508'''  (1992)  pp. 132–148</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Martio,  U. Sebro,  "Universal radius of injectivity for locally quasiconformal mappings"  ''Israel J. Math.'' , '''29'''  (1978)  pp. 17–23</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 19:29, 1 November 2016

In 1967, V.A. Zorich proved the following result for quasi-regular mappings in space: A locally homeomorphic quasi-regular mapping $f : \mathbf{R}^n \rightarrow \mathbf{R}^n$, $n \ge 3$, is, in fact, a homeomorphism of $\mathbf{R}^n$.

This result had been conjectured by M.A. Lavrent'ev in 1938. Note that the exponential function shows that there is no such result for $n=2$. In 1971, O. Martio, S. Rickman and J. Väisälä proved a stronger quantitative result: For $n \ge 3$ and $K > 1$ there exists a number $\psi(n,K) \in (0,1)$ , the radius of injectivity, such that every locally injective $K$-quasi-regular mapping $f : B^n \rightarrow \mathbf{R}^n$, where $B^b = B^n(1)$ and $B^n(r) = \{ x \in \mathbf{R}^n : |x| \le r \}$, for $r > 0$, is injective in $B^n(\psi(n,K))$.

References

[a1] S. Rickman, "Quasiregular mappings" , Ergeb. Math. Grenzgeb. , 26 , Springer (1993)
[a2] V.A. Zorich, "The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems" M. Vuorinen (ed.) , Quasiconformal Space Mappings , Lecture Notes in Mathematics , 1508 (1992) pp. 132–148
[a3] O. Martio, U. Sebro, "Universal radius of injectivity for locally quasiconformal mappings" Israel J. Math. , 29 (1978) pp. 17–23
How to Cite This Entry:
Zorich theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zorich_theorem&oldid=15060
This article was adapted from an original article by M. Vuorinen (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article