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 —
- A probability measure on the Borel sets of is called Gaussian if there exists a pair such that
- Given a probability space and a Borel measurable function , we say that is a Gaussian function if the pushforward is a Gaussian measure.
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 .
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.
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 .