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Difference between revisions of "Aperiodic automorphism"

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012740/a0127401.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012740/a0127402.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012740/a0127403.png" />, 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|ergodic theory]] automorphisms with  "too many"  periodic points are considered as trivial exceptions (see [[#References|[1]]]).
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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>
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<table>
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<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>
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</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. [[Approximation by periodic transformations|Approximation by periodic transformations]]), see [[#References|[a1]]], p. 75, or [[#References|[a2]]], p. 390.
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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>
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<table>
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<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>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,  "Lectures on ergodic theory" , Math. Soc. Japan  (1956)</TD></TR>
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</table>
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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=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