Examples of measure preserving systems with varied behaviours are vital in ergodic theory, to understand the general properties and to have counter examples to false statements. One classical method to craft examples with specific properties is the so-called Gaussian construction. In this post I define and give simple examples of applications of this construction. I took most of the material for this post from the book of Cornfeld, Fomin and Sinai.
— 1. Definition of Gaussian systems —
Definition 1
Using probabilistic terminology, is a random variable,
is the expectation (or mean or first moment) of
and
is the standard deviation. Since by subtracting a constant from
we obtain
, we will from now on assume that every Gaussian function has mean
. The following lemma is well known in probability theory and the proof will not be provided.
Lemma 2 If
is a (real) subspace and every
is a Gaussian function, then any
in the
closure of
is also a Gaussian function.
Next we define Gaussian measure preserving systems.
Definition 3 Let
be a measure preserving system and let
.
- We say that
is Gauss element if any function in the linear span
of
is Gaussian.
- We say that
is a Gaussian system if there exists a Gauss element
in
whose orbit generates the whole
-algebra, i.e.
Such
is called a generating Gaussian element.
We want to stress the obvious fact that being a Gaussian system does not imply that every
is a Gaussian function (consider for instance the indicator function of any measurable set) but rather that it contains a closed invariant subspace
, generating the whole
-algebra, such that every
is a Gaussian function.
— 2. Positive definite sequences and Gaussian systems —
In the same way that a Gaussian measure (with mean ) is uniquely determined by its standard deviation, a Gaussian system is uniquely determined by the correlation sequence
.
It is well known that for any measure preserving system and any
, the sequence
is positive definite:
Definition 4 A function
is called positive definite if for every
with finite support
In particular, the correlation sequence of a Gaussian system is positive definite. It is not so obvious that for every positive definite sequence
there exist a Gaussian system with correlation sequence
, or indeed a measure preserving system
and
with
.
Theorem 5 Every positive definite sequence
is the correlation sequence of a Gaussian system.
While so far we restricted our attention to -actions, Theorem 5 holds in far greater generality. Indeed we can replace the role of
with any group (in fact the result goes even outside the scope of groups, by considering, instead of positive definite sequences, so-called reproducing kernels (which are essentially functions
with domain in the cartesian square of any set
satisfying certain properties. In the case of groups the reproducing kernel takes the shape of
).) The proof of Theorem 5 has two steps, both of which are non-trivial. The first step is to establish a halfway result (it can be found in this formulation as Theorem 5.20 in these notes of Paulsen):
Theorem 6 (Naimark’s Dilation Theorem) For every positive definite sequence
of a group
there exists unitary representation of the group
on a Hilbert space
and a vector
such that
for all
.
Proof: (sketch)
Let be the set of all functions
with finite support. Define a inner product in
by letting
Quotient out the subspace and then let
be the completion of
with respect to the norm
; the extension of the inner product turns
into a Hilbert space. Define
for
as the function with finite support
and extend it by continuity to
; this becomes a unitary representation of
in
. Finally take
to be the indicator function of the singleton containing the identity of
; it is clear that
.
We can now prove Theorem 5:
Proof: Let , let
be the Gaussian measure on
with mean
and standard deviation
and let
be the product measure on the Borel sets of
. Let
and
be the Hilbert space, unitary representation and vector given by Naimark’s dilation theorem. Let
be an orthonormal basis on
and define
by mapping
to
, where
is the projection onto the
-th coordinate. Extend
to
, it follows from Lemma 2 that every function
is a Gaussian function.
Next let , it is a Gaussian element. Since
is an isometric isomorphism, it follows that
, so
is the correlation function of the Gaussian system
.
— 3. Spectral properties of Gaussian systems —
In this section we return to -actions (in fact, most of this section applies to actions of locally compact abelian groups). Let
be a Gaussian system and let
be a generating Gaussian element with correlation sequence
. Bochner-Herglotz theorem states that there exists a measure
on
such that
, where
. The measure
is called the spectral measure of the Gaussian system. Since
it follows that
for any Borel
.
Let denote the smallest closed invariant subspace containing
. One can define the linear map
by sending
to
and extending by linearity and continuity to
. Observe that for a real valued function
, the image under
satisfies
.
Proposition 7 The map
is an isometric isomorphism such that
This proposition, whose proof is routine and will be omitted, already allows us to deduce the first non-trivial fact about Gaussian systems:
Proposition 8 If the spectral measure
has atoms, then the Gaussian system
is not ergodic.
Proof: Let be such that
and let
be the indicator function of the singleton
. Observe that
(to be completely clear, the first member of the equation has a multiplication of functions, the second a multiplication of a scalar with a function). Let
, it follows from Proposition 7 that
Therefore is a
-invariant function. Moreover, since
, both its real and imaginary part are Gaussian functions, therefore
can not be a constant, and this shows that the Gaussian system
is not ergodic.
To show the converse (in a strong sense) to Proposition 8 we need a more fine understanding of the geometry of . Denote by
the one dimensional space of constant functions and let
be the space denote
above. Then, for each
we let
be the orthogonal complement of
in the closed linear span of functions of the form
for any
. Since
generates the full
-algebra, we have the orthogonal decomposition
Observe that each is a closed invariant subspace of
. For any
one can define a map
analogous to the map
defined above. Unfortunately, the construction is significantly more complicated , so I will just state the relevant properties without proof.
Theorem 9 For each
there exists a map
(where
is just the product measure of
with itself
times) satisfying:
- The image of
is the set of functions
which are invariant under permutation of coordinates.
- The map
is an isometric isomorphism between
and its image.
- Let
denote the map
. Then for every
![]()
We can now prove the converse of Proposition 8.
Proposition 10 Let
be a Gaussian system . If the spectral measure
has no atoms, then the system
is weak mixing.
Proof: Let be an eigenvector satisfying
; we need to show that it is constant. Using (2), decompose
. Since the spaces
are orthogonal and invariant, each
is itself an eigenvector with
. Next, fix
and let
. Using Theorem 9 we deduce that
in
. More precisely
Since is non-atomic, the codimension
shifted subtorus
has measure
. Hence it follows from (3) that
. Since
was arbitrary, we conclude that
, i.e. it is a constant as desired.
Corollary 11 A Gaussian system is weak mixing if and only if it is ergodic.
Another property which can be easily deduced from the decomposition (2) characterizes strong mixing of a Gaussian system in terms of properties of the spectral measure .
Proposition 12 Let
be a Gaussian system. The system is strongly mixing if and only if
.
Proof: Let be a generating Gaussian element. If the system is strongly mixing then
.
Now assume that and let
for some
. Let
be the pullback
-algebra of the Borel
-algebra on
by the map
. Let
, let
be the convolution of
with itself
times (equivalently,
is the pushforward of
through the map
) and let
be the conditional expectation of
on
. We have
Since , we have that
. Therefore the generalized Riemann-Lebesgue lemma implies that
, finishing the proof.
It is a known fact that there exist continuous singular measures on
such that
does not go to
as
. This implies that there exist (Gaussian) systems which are weak mixing but not strongly mixing.
— 4. Rigid -action mixing of all shapes —
One interesting construction using Gaussian systems is of a rigid -action mixing of all shapes.
Definition 13 Let
be an abelian group and
be a finite set.
- A probability preserving action
of
is
-mixing if
for any
for
.
- The system
is mixing of all shapes if it is
-mixing for every finite
.
- The system
is rigid if there exists a sequence
in
such that
for all
.
Observe that if a system is mixing of order , then it is
-mixing for every
with
. However, the converse is not necessarily true.
Theorem 14 (Ferenczi and Kamiński) There exists a probability preserving action of
which is rigid and mixing of all shapes.
Proof: (Sketch)
In view of Theorem 5, it suffices to construct a positive definite sequence with certain properties. Indeed one can show that if is a positive definite sequence such that
for all
with
, then the Gaussian system induced by
is mixing of all shapes. Similarly, if
and there exists a sequence
in
along which
, then the induced Gaussian system is rigid (exactly along
). Taking
linearly independent over
, the sequence
satisfies the conditions. Indeed, it is clear to show that for any
with
. It is also not hard to find a sequence
in
such that
, which implies that
.