# Aperiodic automorphism

From Encyclopedia of Mathematics

*of a measure space*

An automorphism $T$ of a measure space such that its periodic points, i.e. the points $x$ for which $T^k(x) = x$ for some $k>0$, form a set of measure zero. The introduction of a special name for such transformations is due to the fact that in certain theorems of ergodic theory automorphisms with "too many" periodic points are considered as trivial exceptions (see [1]).

#### References

[1] | V.A. Rokhlin, "Selected topics from the metric theory of dynamical systems" Amer. Math. Soc. Transl. Series 2 , 49 pp. 171–240 Uspekhi Mat. Nauk , 4 : 2 (30) (1949) pp. 57–128 |

#### Comments

The so-called Rokhlin–Halmos lemma for periodic automorphisms is important for the approximation of an automorphism of a Lebesgue space by periodic transformations (cf. Approximation by periodic transformations), see [a1], p. 75, or [a2], p. 390.

#### References

[a1] | I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian) |

[a2] | P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) |

**How to Cite This Entry:**

Aperiodic automorphism.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Aperiodic_automorphism&oldid=39716

This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article