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Difference between revisions of "Integral automorphism"

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The same as a [[Special automorphism|special automorphism]], constructed from an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i0513501.png" /> of a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i0513502.png" /> and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i0513503.png" /> (given on this space and taking values in the positive integers). The term  "integral automorphism"  is mostly used in the non-Soviet literature.
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The same as a [[Special automorphism|special automorphism]], constructed from an automorphism  $  T $
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of a [[Measure space|measure space]]  $  ( X , \mu ) $
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and a function  $  F $(
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given on this space and taking values in the positive integers). The term  "integral automorphism"  is mostly used in the non-Soviet literature.
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i0513504.png" /> be the measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i0513505.png" /> with measure
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Let $  X  ^ {F} $
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be the measure space $  X  ^ {F} = \{ ( x , i ) \in X \times \mathbf N \cup \{ 0 \} : 0 \leq  i < F ( x) \} $
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with measure
  
<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/i/i051/i051350/i0513506.png" /></td> </tr></table>
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$$
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\mu  ^ {F} ( A)  =
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\frac{\mu ( A) }{\int\limits _ {x} F ( x)  d \mu }
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.
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$$
  
Then the integral automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i0513507.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i0513508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i0513509.png" /> is the automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i05135010.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i05135011.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i05135012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i05135013.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051350/i05135014.png" />. For more details see [[#References|[a1]]] and [[Special automorphism|Special automorphism]].
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Then the integral automorphism $  T  ^ {F} $
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corresponding to $  T $
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and $  F $
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is the automorphism of $  X  ^ {F} $
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defined by $  T  ^ {F} ( x , i ) = ( x , i + 1 ) $
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if $  i + 1 < F ( x) $,  
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and $  T  ^ {F} ( x , i ) = ( T x , 1 ) $
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if  $  i + 1 = F ( x) $.  
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For more details see [[#References|[a1]]] and [[Special automorphism|Special automorphism]].
  
 
====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)  pp. Chapt. 1, Sect. 5  (Translated from Russian)</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)  pp. Chapt. 1, Sect. 5  (Translated from Russian)</TD></TR></table>

Latest revision as of 22:12, 5 June 2020


The same as a special automorphism, constructed from an automorphism $ T $ of a measure space $ ( X , \mu ) $ and a function $ F $( given on this space and taking values in the positive integers). The term "integral automorphism" is mostly used in the non-Soviet literature.

Comments

Let $ X ^ {F} $ be the measure space $ X ^ {F} = \{ ( x , i ) \in X \times \mathbf N \cup \{ 0 \} : 0 \leq i < F ( x) \} $ with measure

$$ \mu ^ {F} ( A) = \frac{\mu ( A) }{\int\limits _ {x} F ( x) d \mu } . $$

Then the integral automorphism $ T ^ {F} $ corresponding to $ T $ and $ F $ is the automorphism of $ X ^ {F} $ defined by $ T ^ {F} ( x , i ) = ( x , i + 1 ) $ if $ i + 1 < F ( x) $, and $ T ^ {F} ( x , i ) = ( T x , 1 ) $ if $ i + 1 = F ( x) $. For more details see [a1] and Special automorphism.

References

[a1] I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) pp. Chapt. 1, Sect. 5 (Translated from Russian)
How to Cite This Entry:
Integral automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_automorphism&oldid=47365
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article