Difference between revisions of "Integral automorphism"
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| + | $#C+1 = 14 : ~/encyclopedia/old_files/data/I051/I.0501350 Integral automorphism | ||
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| + | The same as a [[Special automorphism|special automorphism]], constructed from an automorphism $ T $ | ||
| + | of a [[Measure space|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==== | ====Comments==== | ||
| − | Let | + | 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 | + | 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 [[#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
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