In mathematics, the Rokhlin lemma, or Kakutani–Rokhlin lemma is an important result in ergodic theory. It states that an aperiodic measure preserving dynamical system can be decomposed to an arbitrary high tower of measurable sets and a remainder of arbitrarily small measure. It was proven by Vladimir Abramovich Rokhlin and independently by Shizuo Kakutani. The lemma is used extensively in ergodic theory, for example in Ornstein theory and has many generalizations.
Rokhlin lemma belongs to the group mathematical statements such as Zorn's lemma in set theory and Schwarz lemma in complex analysis which are traditionally called lemmas despite the fact that their roles in their respective fields are fundamental.
Lemma: Let
T
:
X
→
X
be an invertible measure-preserving transformation on a standard measure space
(
X
,
Σ
,
μ
)
with
μ
(
X
)
=
1
. We assume
T
is aperiodic, that is, the set of periodic points for
T
has zero measure. Then for every integer
n
∈
N
and for every
ε
>
0
, there exists a measurable set
E
such that the sets
E
,
T
E
,
…
,
T
n
−
1
E
are pairwise disjoint and such that
μ
(
E
∪
T
E
∪
⋯
∪
T
n
−
1
E
)
>
1
−
ε
A useful strengthening of the lemma states that given a finite measurable partition
P
, then
E
may be chosen in such a way that
T
i
E
and
P
are independent for all
0
≤
i
<
n
.
Let
(
X
,
T
)
be a topological dynamical system consisting of a compact metric space
X
and a homeomorphism
T
:
X
→
X
. The topological dynamical system
(
X
,
T
)
is called minimal if it has no proper non-empty closed
T
-invariant subsets. A topological dynamical system
(
Y
,
S
)
is called a factor of
(
X
,
T
)
if there exists a continuous surjective mapping
φ
:
X
→
Y
which is equivariant, i.e.,
φ
(
T
x
)
=
S
φ
(
x
)
for all
x
∈
X
.
Lindenstrauss proved the following theorem:
Theorem: Let
(
X
,
T
)
be a topological dynamical system which has an aperiodic minimal factor. Then for integer
n
∈
N
there is a continuous function
f
:
X
→
R
such that the set
E
=
{
x
∈
X
∣
f
(
T
x
)
≠
f
(
x
)
+
1
}
satisfies
E
,
T
E
,
…
,
T
n
−
1
E
are pairwise disjoint.
There is a version for non-invertible measure preserving systems.
Ornstein and Weiss proved a version for free actions by countable discrete amenable groups.
C. Linderholm proved a version for periodic non-singular transformations.
