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)