# Expanding mapping

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