View source for Derived automorphism

<!--
d0312701.png
$#A+1 = 29 n = 0
$#C+1 = 29 : ~/encyclopedia/old_files/data/D031/D.0301270 Derived automorphism
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''in ergodic theory''

A transformation  $  T _ {X} $
defined by using an automorphism  $  T $
of a [[Measure space|measure space]]  $  ( M , \mu ) $
and a measurable subset  $  X \subset  M $
of positive measure such that almost-all points of  $  X $
return to  $  X $
under the action of iterates of  $  T $. 
For every such point  $  x $
its image  $  T _ {X} ( x) $
is defined as that point of the trajectory  $  T  ^ {n} x $
at which this trajectory returns to  $  X $
for the first time after  $  x $(
according to the Poincaré recurrence theorem, cf. [[Poincaré return theorem|Poincaré return theorem]], the condition for almost-all points of  $  X $
to return to  $  X $
at some time is automatically satisfied if  $  \mu ( M) < \infty $). 
The transformation  $  T _ {X} $
turns out to be an automorphism (more precisely, an automorphism modulo  $  0 $) 
of the space  $  X $
with the measure induced on it (this measure is the measure  $  \mu $
considered only on subsets of  $  X $; 
if  $  \mu ( X) < \infty $
then this measure is usually normalized).

Conversely, if  $  \cup _ {n \geq  0 }  T  ^ {n} X = M $(
this condition is automatically satisfied if the automorphism  $  T $
is ergodic), then the original automorphism  $  T $
can be recovered (up to conjugation by means of an isomorphism of measure spaces) from  $  T _ {X} $
and the time of first return

$$ 
n _ {X} ( x)  =  \min \{ {n > 0 } : {T  ^ {n} x \in X } \}
.
$$

Namely,  $  T $
is the [[Special automorphism|special automorphism]] constructed from  $  T _ {X} $
and  $  n _ {X} $.

====Comments====
For automorphism of a measure space cf. [[Measure-preserving transformation|Measure-preserving transformation]].

In the literature also induced or derivative automorphism are used. See [[#References|[a1]]] or [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kakutani,   "Induced measure preserving transformations"  ''Proc. Japan. Acad.'' , '''19'''  (1943)  pp. 635–641</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Petersen,   "Ergodic theory" , Cambridge Univ. Press  (1983)  pp. 39</TD></TR></table>