Special automorphism
constructed from an automorphism of a measure space
and a function
(defined on
and taking positive integral values)
An automorphism of a certain new measure space
constructed in the following way. The points of
are the pairs
where
and
is an integer,
, and
is equipped with the obvious measure
: if
and
for all
, then
. If
, then one usually normalizes this measure. Let
be the transformation that increases the second coordinate of the point
by one if
(i.e. if the transformed point remains within
), and otherwise put
. The transformation
turns out to be an automorphism of the measure space
.
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 with
, one may assume that
. Then
is the time spent by a point that starts in
and moves under the action of the cascade
to return once again to
, and
is the induced automorphism
. 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
.
Comments
Instead of "special automorphism constructed from an automorphism S" one also speaks of a primitive of . (In that case what was called above the "induced automorphism" is called a derivative of
. See [a2].) The idea goes back to S. Kakutani; cf. [a1].
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 |
[a3] | I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) pp. Chapt. 1, §5 (Translated from Russian) |
Special automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Special_automorphism&oldid=15242