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Difference between revisions of "Expanding mapping"

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A differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e0368701.png" /> of a closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e0368702.png" /> onto itself, under the action of which the lengths of all tangent vectors (in the sense of some, and thus of any, Riemannian metric) grow at an exponential rate, that is, there are constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e0368703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e0368704.png" />, such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e0368705.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e0368706.png" />,
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A differentiable mapping $f$ of a closed manifold $M$ onto itself, under the action of which the lengths of all tangent vectors (in the sense of some, and thus of any, Riemannian metric) grow at an exponential rate, that is, there are constants $C>0$ and $\lambda>1$, such that for all $X\in TM$ and all $n>0$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e0368707.png" /></td> </tr></table>
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$$\|Tf^n(X)\|\geq C\lambda^n\|X\|.$$
  
There is also a variant of this concept without the condition of differentiability, covering as special cases many previously studied one-dimensional examples. The properties of expanding mappings are analogous to those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e0368708.png" />-systems (cf. [[Y-system|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e0368709.png" />-system]]) and in part are even simpler (thus, an expanding mapping of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e03687010.png" /> always has a finite invariant measure, defined in terms of the local coordinates as a positive density).
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There is also a variant of this concept without the condition of differentiability, covering as special cases many previously studied one-dimensional examples. The properties of expanding mappings are analogous to those of $Y$-systems (cf. [[Y-system|$Y$-system]]) and in part are even simpler (thus, an expanding mapping of class $C^2$ always has a finite invariant measure, defined in terms of the local coordinates as a positive density).
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036870/e03687011.png" />-system is usually called an Anosov system in English.
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A $Y$-system is usually called an Anosov system in English.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Shub,  "Expanding maps"  S.-S. Chern (ed.)  S. Smale (ed.) , ''Global analysis'' , ''Proc. Symp. Pure Math.'' , '''14''' , Amer. Math. Soc.  (1970)  pp. 273–276</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Shub,  "Expanding maps"  S.-S. Chern (ed.)  S. Smale (ed.) , ''Global analysis'' , ''Proc. Symp. Pure Math.'' , '''14''' , Amer. Math. Soc.  (1970)  pp. 273–276</TD></TR></table>

Latest revision as of 19:13, 9 October 2014

A differentiable mapping $f$ of a closed manifold $M$ onto itself, under the action of which the lengths of all tangent vectors (in the sense of some, and thus of any, Riemannian metric) grow at an exponential rate, that is, there are constants $C>0$ and $\lambda>1$, such that for all $X\in TM$ and all $n>0$,

$$\|Tf^n(X)\|\geq C\lambda^n\|X\|.$$

There is also a variant of this concept without the condition of differentiability, covering as special cases many previously studied one-dimensional examples. The properties of expanding mappings are analogous to those of $Y$-systems (cf. $Y$-system) and in part are even simpler (thus, an expanding mapping of class $C^2$ always has a finite invariant measure, defined in terms of the local coordinates as a positive density).

References

[1] M. Shub, "Endomorphisms of compact differentiable manifolds" Amer. J. Math. , 91 : 1 (1969) pp. 175–199
[2] P. Walters, "Invariant measures and equilibrium states for some mappings which expand distances" Trans. Amer. Math. Soc. , 236 (1978) pp. 121–153
[3] K. Krzyzewski, "A remark on expanding mappings" Colloq. Math. , 41 : 2 (1979) pp. 291–295
[4] K. Krzyzewski, "Some results on expanding mappings" Astérisque , 50 (1977) pp. 205–218
[5] K. Krzyzewski, "On analytic invariant measures for expanding mappings" Colloq. Math. , 46 : 1 (1982) pp. 56–65
[6] M. Gromov, "Groups of polynomial growth and expanding maps" Publ. Math. IHES , 53 (1981) pp. 53–78


Comments

A $Y$-system is usually called an Anosov system in English.

References

[a1] M. Shub, "Expanding maps" S.-S. Chern (ed.) S. Smale (ed.) , Global analysis , Proc. Symp. Pure Math. , 14 , Amer. Math. Soc. (1970) pp. 273–276
How to Cite This Entry:
Expanding mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Expanding_mapping&oldid=33519
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article