Derived automorphism

in ergodic theory

A transformation $ T _ {X} $ defined by using an automorphism $ T $ of a 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, 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 constructed from $ T _ {X} $ and $ n _ {X} $.

Comments

For automorphism of a measure space cf. Measure-preserving transformation.

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

References

[a1]	S. Kakutani, "Induced measure preserving transformations" Proc. Japan. Acad. , 19 (1943) pp. 635–641
[a2]	K. Petersen, "Ergodic theory" , Cambridge Univ. Press (1983) pp. 39