# Special automorphism

constructed from an automorphism $S$ of a measure space $(X,\nu)$ and a function $f$ (defined on $X$ and taking positive integral values)

An automorphism $T$ of a certain new measure space $(M,\mu)$ constructed in the following way. The points of $M$ are the pairs $(x,n)$ where $x\in X$ and $n$ is an integer, $0\leq n<f(x)$, and $M$ is equipped with the obvious measure $\mu$: if $A\subset X$ and $f(x)>n$ for all $x\in A$, then $\mu(A\times\{n\})=\nu(A)$. If $\mu(M)=\int_Xfd\nu<\infty$, then one usually normalizes this measure. Let $T$ be the transformation that increases the second coordinate of the point $(x,n)$ by one if $n+1<f(x)$ (i.e. if the transformed point remains within $M$), and otherwise put $T(x,n)=(Sx,0)$. The transformation $T$ turns out to be an automorphism of the measure space $(M,\mu)$.

The above construction is often applied in ergodic theory when constructing various examples. On the other hand, the role of special automorphisms is clear from the following. By identifying each point $x\in X$ with $(x,0)$, one may assume that $X\subset M$. Then $f(x)$ is the time spent by a point that starts in $X$ and moves under the action of the cascade $\{T^n\}$ to return once again to $X$, and $S$ is the induced automorphism $T_X$. Thus, special automorphisms can be used to recover the trajectories of a dynamical system in the whole phase space by observing only the passages of the moving point through the set $X$.

Instead of "special automorphism constructed from an automorphism S" one also speaks of a primitive of $S$. (In that case what was called above the "induced automorphism" is called a derivative of $S$. See [a2].) The idea goes back to S. Kakutani; cf. [a1].