Integral automorphism

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)