Difference between revisions of "Aperiodic automorphism"
From Encyclopedia of Mathematics
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''of a measure space'' | ''of a measure space'' | ||
| − | An automorphism T of a measure space such that its periodic points, i.e. the points | + | 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 [[#References|[1]]]). |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> 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</TD></TR> | ||
| + | </table> | ||
====Comments==== | ====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. [[ | + | 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 [[#References|[a1]]], p. 75, or [[#References|[a2]]], p. 390. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian)</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Revision as of 19:33, 9 November 2016
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=16898
Aperiodic automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aperiodic_automorphism&oldid=16898
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article