Lecture notes on ergodic transformations

Valery V. Ryzhikov

Our aim is to introduce the reader to some classical properties of ergodic
transformations and some problems (many of them are not so simple). The text
is based on lectures that the author gave at the Faculty of Mechanics and
Mathematics of Moscow State University for senior students wishing to become
familiar with the theory of ergodic transformations. Several introductory
lectures were held at the international summer school at Moscow State
University on July 23-25, 2024, what was the reason for publishing this
version.

Content

§1. Introduction.

§2. Properties of transformations equivalent to ergodicity.

§3. Birkhoff’s Theorem.

§4. Properties equivalent to weak mixing.

§5. On typical properties of transformations.

§6. Lego to constuct transformations.

§7. Typical entropy invariants.

§8. Poisson suspensions with completely positive $P$-entropy.

§9. Spectral theorem for unitary operators.

§10. Compact factors, Kronecker algebra.

§11. Progression recurrence for weakly mixing transformations.

§12. Double recurrence for ergodic transformations.

##  1 Introduction

For brevity, by transformation we mean a measure-preserving invertible
transformation $T$ of the Lebesgue space $(X,{\mathcal{B}},\mu)$. Such
transformations are often called automorphisms of a measure space. Now we will
discuss the standard probability space, i.e. $\mu(X)=1$. Transformations that
differ on a set of measure 0 are identified.

Ergodicity. Transformation $T$ is ergodic if every measurable set $A$ such
that $A=TA$, has measure 0 or 1. In other words, the phase space $X$ is
indecomposable: cannot be "cut" into two nontrivial invariant parts.

Weak mixing. Transformation $T$ is called weakly mixing if there is a sequence
$n_{i}\to\infty$ such that

$\forall A,B\in{\mathcal{B}}\ \ \ \mu(T^{n_{i}}A\cap B)\to\mu(A)\mu(B),\
i\to\infty.$

Mixing. Transformation $T$ is called mixing if

$\forall A,B\in{\mathcal{B}}\ \ \ \mu(T^{i}A\cap B)\to\mu(A)\mu(B),\
i\to\infty.$

(This property is not typical in the Baire sense, we will prove this later.)

Example of ergodic transformation. Let $a$ be a real number, consider the
shift $T_{a}\ :[0,1)\to[0,1)$

$T_{a}(x)=\\{x+a\\}:=x+a\ (mod\ 1).$

The transformation $T_{a}$ is ergodic with respect to Lebesgue measure on
$[0,1)$ iff $a$ is irrational. Hint: find an elementary proof of this fact
based on the fact that any (!) measurable set $A$ of positive measure
intersects some segment of measure ${\varepsilon}$ to a measure greater than
$0.9{\varepsilon}$. If $TA=A$, then can get $\mu(A)>0.8$, hence, $\mu(A)=1$ or
$\mu(X\setminus A)>0.8$ (the latter is impossible since $\mu(A)>0.8$).

Let’s consider a more general example: a group shift on a torus

$T_{(a_{1},\dots,a_{n})}(x_{1},\dots,x_{n})=(\\{x_{1}+a_{1}\\},\dots,\\{x_{n}+a_{n}\\}),$

where $\\{x+a\\}$ denotes the fractional part of $x+a$. What conditions should
be imposed on $a_{1},\dots,a_{n}$ so that the shift is ergodic with respect to
the Lebesgue measure on the torus?

An example of a mixing transformation. Consider the Bernoulli shift
$T:{\mathbf{Z}}_{2}^{{\mathbf{Z}}}\to{\mathbf{Z}}_{2}^{{\mathbf{Z}}}$ on the
space of two-sided sequences taking the values from $\\{0,1\\}$. For $x$ such
a sequence, we consider the shift

$T(x)_{z}=x_{z-1}.$

The measure $\mu$ on $X={{\mathbf{Z}}}_{2}^{{\mathbf{Z}}}$ is defined as
follows: the set of all sequences with fixed values for on a finite set $F$ of
indices $z$ has measure $\frac{1}{2^{|F|}}$. It is natural to call such sets
cylinders; they form a semiring on which we have defined a countably additive
measure called Bernoulli mesure of type
$\left(\frac{1}{2},\frac{1}{2}\right)$. It is sufficient to verify the mixing
property on cylinders. If $A$ and $B$ are cylinders, then

$\mu(T^{i}A\cap B)=\mu(A)\mu(B)$

for all sufficiently large $i$, which follows easily from the definition of
the measure.

Note that the shift $T$ is an automorphism of the compact group
${{\mathbf{Z}}}_{2}^{{\mathbf{Z}}}$ that preserves its Haar measure (here,
this is a probability measure invariant with respect to all group shifts). And
this Bernoulli-Haar measure is also Lebesgue on a square. Indeed, the
automorhism $T$ is also called the baker’s transformation of $X$ (as a
square), and $T$ maps a point to another point of the square as follows:

$T\left(\sum_{i=1}^{\infty}\frac{x_{i}}{2^{i}},\
\sum_{i=1}^{\infty}\frac{x_{-i+1}}{2^{i}}\right)=\
\left(\sum_{i=1}^{\infty}\frac{x_{i-1}}{2^{i}},\
\sum_{i=1}^{\infty}\frac{x_{-i}}{2^{i}}\right).$

The baker’s transformation $T:[0,1]^{2}\to[0,1]^{2}$ preserves Lebesgue
measure on a square $[0,1]^{2}$. Locally, it contracts along one coordinate
and expands along the other.

The property of weak mixing follows directly from the property of mixing,
which implies ergodicity. Indeed, if $TA=A$, then $T^{n}A=A$, then taking into
account weak mixing for $B=A$, we have

$\mu(A)=\mu(T^{n_{i}}A\cap A)\to\mu(A)\mu(A),\ \ \ \mu(A)\in\\{0,1\\}.$

This means that $T$ is ergodic (the invariant set is all or nothing).

Note that circle rotations (shifts on tori) do not have weak mixing. Weak
mixing implies the following property: the sets $T^{n_{i}}A$ asymptotically
have a nonempty intersection with any fixed sets of positive measure. However,
a small segment under rotation cannot intersect two distant segments
simultaneously – the absence of the mixing in this case is obvious.

Koopman operators. Let us consider for an invertible measure-preserving
transformation $T$ a unitary operator $T$ acting in
$L_{2}(X,{\mathcal{B}},\mu)$ according to the simple formula

$Tf(x)=f(T(x)).$

We denote the operator and the transformation in the same way, this does not
lead to misunderstandings. In what follows, we write $Tx$ instead of $T(x)$.
The mentioned shifts by ${{\mathbf{Z}}}_{2}^{{\mathbf{Z}}}$ and by
${{\mathbf{Z}}}_{3}^{{\mathbf{Z}}}$ are isomorphic as operators: they are
identically structured permutations of suitable orthonormal systems. If the
reader is familiar with the characters of commutative groups (or with
Rademacher-Walsh systems), then he will quickly find these systems and shifts
on them. However, as Kolmogorov showed, unitary isomorphism is not induced by
any measure-preserving transformation. Two transformations $S$, $T$ are called
metrically isomorphic if there is a transformation $R$ such that $S=R^{-1}TR$.

Denote by $\Theta$ the orthoprojection onto the space of constants in $L_{2}$.
The mixing property is equivalent to the convergence

$T^{i}\to_{w}\Theta\ :\ \ \int_{X}T^{i}f\,g\,d\mu\ \to\
\int_{X}f\,d\mu\int_{X}g\,d\mu,\ i\to\infty,$

for all $f,g\in L_{2}(X,\mu)$. If the operators $T^{n_{i}}$ converge weakly to
$\Theta$:

$T^{n_{i}}\to_{w}\Theta$

for some sequence ${n_{i}}$, we call such $T$ weakly mixing.

The ergodicity property can also be formulated in terms of weak limits:

$\frac{1}{N}\sum_{1}^{N}T^{i}\to_{w}\Theta.$

We will verify this later.

Infinite family of non-isomorphic weakly mixing non-mixing transformations. If
we are familiar with the ergodic rotation of the circle and the Bernoulli
shift, and also know the Rokhlin-Halmos lemma, then in fact we are able to
construct an uncountable family of non-isomorphic transformations. Let’s try.

Rokhlin-Halmos lemma. For an ergodic transformation $T$, ${\varepsilon}>0$ and
a natural $H$, there exists a measurable set $B$ such that the sets $T^{i}B$
are disjoint for $0\leq i\leq H-1$ and

$\mu\left({\bigsqcup}_{i=0}^{H-1}T^{i}B\right)>1-{\varepsilon}.$

Exercises. Find a set $A$ such that for some $n\gg H$ we have

$\mu\left({\bigsqcup}_{i=0}^{n}T^{i}A\right)>0.$

– Represent the entire space $X$ as a union of disjoint (high) towers:

$X={\bigsqcup}_{n>N}{\bigsqcup}_{i=0}^{n}T^{i}A_{n},\ \ N\gg H,$

and prove the lemma.

– Prove the Lehrer-Weiss $\varepsilon$-free lemma:

If $T^{n}$, $n>0$, is ergodic, then for any set $A$, $\mu(A)>0$, there exists
a measurable set $B$ such that

$X\setminus A\ \subset\ {\bigsqcup}_{i=0}^{n-1}T^{i}B.$

Corollary of the Rokhlin-Halmos lemma. For ergodic transformations
$T,T^{\prime}$ and any $\delta>0$, one can modify the transformation $T$ on a
set of measure less than $\delta$, such that the modified transformation is
isomorphic to (conjugate to) $T^{\prime}$.

Hint: For $T$ and $T^{\prime}$ we apply R-H lemma with
$\mu(B)=\mu(B^{\prime})$ (this is easy to ensure) and ${\varepsilon}$ and $H$
such that ${\varepsilon}+\frac{1}{H}<<\delta$. Next we find $R$ such that on
the set ${\bigsqcup}_{i=0}^{H-2}{T^{\prime}}^{i}B^{\prime}$ the conjugation
$R^{-1}TR$ coincides with $T^{\prime}$.

Weakly mixing rigid transformations. An ergodic rotation of the circle $S$ has
the rigidity property: $S^{k(j)}\to_{w}I$, where $I$ is the identity operator,
$k(j)\to\infty$ (note that weak convergence of unitary operators to a unitary
operator turns out to be strong convergence). The rigidity property, like the
mixing property, is invariant under conjugation. We will use this now.

Consider a sequence of transformations $T_{p}$ such that

$\sum_{p}\mu(\\{x:T_{p}(x)\neq T_{p+1}\\})<\infty,$

then $T_{p}$ converge almost everywhere to the transformation $T$. We will
choose $T_{2p}$ isomorphic to the rigid transformation, and $T_{2p+1}$
isomorphic to the mixing transformation, Since the rate of convergence of the
series is allowed to be chosen arbitrarily fast, for the limit transformation
$T$ we can ensure convergence

$T_{2p+1}^{k(2p+1)}\approx_{w}T^{k(2p+1)}\to_{w}\Theta,\ \ \ \ \
T_{2p}^{k(2p)}\approx_{w}T^{k(2p)}\to_{w}I,\ p\to\infty.$

The above notation $T_{2p+1}^{k(2p+1)}\approx_{w}T^{k(2p+1)}$ means that
$T_{2p+1}^{k(2p+1)}-T^{k(2p+1)}\to_{w}0$.

Thus, by varying the transformation on sets of very small measure, we can
control the rigid and mixing sequences and thus obtain a continuum of non-
isomorphic weakly mixing but non-mixing transformations. We recall that for
isomorphic transformations, both rigid and mixing sequences coincide.

Amusing problems.

–Prove that the circle rotation and the Bernoulli shift are metrically
isomorphic to their inverses, respectively.

– The full group $[T]$ of $T$ consists of all so-called generalized powers,
that is, $S\in[T]$ means that the invertible transformation $S$ has the form
$S(x)=T^{n(x)}x$, where $n(x)$ is some measurable function taking integer
values. (The famous Dye’s theorem states that the full group of an ergodic
transformation contains isomorphic copies of all ergodic transformations!)
Prove that an ergodic transformation $T$ is not conjugate to its inverse in
the group $[T]$.

– Following Arnaldo Nogueira, we rotate the circle by some angle. Then on some
arc of this circle we make the following flip:
$\alpha+\varphi\to\alpha-\varphi$, where $\alpha$ is the center of the arc (do
nothing outside the arc). As a result of the above composition we get a
transformation of the circle. Prove that this transformation is periodic (this
surprises at times).

– Let’s consider now the composition of an ergodic group shift $T$ on the
2-torus and a rotation $S$ by 180 degrees of some small disk in the torus.
Outside the disk, the transformation $S$ is identical. Prove that the
composition $ST$ is periodic.

– Now we consider the composition $ST$ of the baker transformation
$T:[0,1]^{2}\to[0,1]^{2}$ with the flip $S:[0,1]^{2}\to[0,1]^{2}$, which is
the symmetry of a triangle with vertices $(0,0),(\varepsilon,0),$
$(0,\varepsilon)$ relative to the diagonal in a square ($\varepsilon>0$).
Outside the triangle $S=Id$. Prove that the Lyapunov exponents (see Valery
Oseledets multiplicative ergodic theorem) for $ST$ are zero. If this seems
complicated, prove simply that almost all points of the square are periodic
with respect to $ST$.

##  2 Properties of transformations equivalent to ergodicity

The ergodicity of a transformation $T$ with respect to a measure $\mu$ is
equivalent to the following properties.

1\. If $f$ is $\mu$-measurable and $Tf=f$, then $f$ is a constant almost
everywhere.

2\. If the measure $\nu\ll\mu$ and $\nu$ is invariant under $T$, then
$\nu=c\,\mu$, $c\geq 0$.

3\. Average mixing: for any measurable $A,B$

$\frac{1}{N}\sum_{1}^{N}\mu(T^{i}A\cap B)\to\mu(A)\mu(B),$

which is equivalent to the weak convergence

$\frac{1}{N}\sum_{1}^{N}T^{i}\to_{w}\Theta.$

4\. Von Neumann’s theorem on strong convergence of ergodic averages:

$\frac{1}{N}\sum_{1}^{N}T^{i}\to_{s}\Theta.$

5\. Birkhoff’s ergodic theorem (convergence almost everywhere).

Let $f\in L_{1}$ and $T$ be an ergodic transformation. Then for almost all $x$
we have the convergence

$\frac{1}{N}\sum_{1}^{N}f(T^{i}x)\to\int fd\mu.$

(Professionals say so: for an ergodic dynamical system the temporal averages
converge to the spatial average.)

(1) The first assertion follows from the fact that $\mu(\\{x:f(x)>c\\})$ is 0
or 1 for all $c$, hence $f=const=\inf$ over all $a$ for which
$\mu(\\{x:f(x)>a\\})=0$.

(2) The second assertion. The Radon-Nikodym derivative (in other words, the
density) of the measure $\nu$ with respect to $\mu$ is invariant with respect
to $T$, therefore, it is constant (here we applied the first assertion). We
obtain $\nu=c\mu$, where $c$ is a non-negative number.

(3) Let us prove the average mixing. Let us fix a measurable set $B$. From any
sequence, one can select a subsequence $N_{m}$ such that for any $A$ the
convergence

$s_{m}(A)=\frac{1}{N_{m}}\sum_{i=1}^{N_{m}}\mu(T^{i}A\cap B)\to\nu(A),$

where $\nu$ is some measure (by the way, why is that?).

By averaging $\nu(A)=\nu(TA)$, indeed,

$|s_{m}(TA)-s_{m}(A)|\leq\frac{2}{N_{m}},$

therefore $s_{m}(TA)\to\nu(A)$ and $s_{m}(TA)\to\nu(TA)$. Thus, the measure
$\nu$ is invariant with respect to $T$.

The measure $\nu$ is absolutely continuous with respect to $\mu$. This is
evident from the equality

$\mu=\nu+\nu^{\prime},$

where $\nu^{\prime}$ is defined in the same way as $\nu$, but instead of $B$
we consider $B^{\prime}=X\setminus B$.

Thus, by virtue of assertion 2, we obtain $\nu=const\,\mu=\mu(B)\mu$. The
second equality is obtained if we substitute $X$ for $A$. We have

$\frac{1}{N_{m}}\sum_{i=1}^{N_{m}}\mu(T^{i}A\cap B)\to\mu(A)\mu(B),$

and since the sequence $N_{m}$ was chosen as a subsequence of an arbitrary
sequence, we have

$\frac{1}{N}\sum_{i=1}^{N}\mu(T^{i}A\cap B)\to\mu(A)\mu(B),\ \ N\to\infty.$

We present an operator proof of a similar fact: If $T$ is an ergodic
transformation, then

$\frac{1}{N}\sum_{1}^{N}T^{i}\to_{w}\Theta.$

We use the fact that any norm-bounded family of operators in $L_{2}$ has a
limit point in the weak operator topology. Let
$\frac{1}{N_{m}}\sum_{1}^{N_{m}}T^{i}\to_{w}P$. For any $f\in L_{2}$, in view
of averaging, we have $TPf=Pf$, therefore, by virtue of the first assertion,
$Pf=const=\int fd\mu$ (prove the second equality). This means that $P=\Theta$.

(4) We prove the strong convergence of

$\frac{1}{N}\sum_{1}^{N}T^{i}\to_{s}0,$

restricting ourselves to the space of functions with zero mean.

The strong convergence of $P_{j}\to_{s}0$ means that
$P_{j}^{\ast}P_{j}\to_{w}0$. Indeed,

$\|P_{j}f\|^{2}=(P_{j}f,P_{j}f)=(P_{j}^{\ast}P_{j}f,f)\to 0,\ \ \|P_{j}f\|\to
0.$

For $P_{N}=\frac{1}{N}\sum_{1}^{N}T^{i}$ we have

$\|TP_{N}^{\ast}P_{N}-P_{N}^{\ast}P_{N}\|\to 0,$

therefore the limit operator $Q$ for the sequence $P_{N_{i}}$ satisfies the
condition

$TQ=Q.$

Due to the ergodicity of $T$ we obtain $Q=0$, for the function $f$ with zero
mean we have

$Qf=TQf=const=0.$

We obtain

$P_{N}^{\ast}P_{N}\to 0,\ \ \|P_{N}f\|\to 0,$

as required.

##  3 Birkhoff’s Theorem

Theorem 3.1. Let $f\in L_{1}$ and $T$ be an ergodic automorphism of a
probability space. Then for almost all $x$ we have the convergence

$\frac{1}{N}\sum_{1}^{N}f(T^{i}x)\to\int fd\mu.$

It suffices to consider the case $\int fd\mu=0$, since the general case is
trivially reducible to it. Let

$X_{a}=\left\\{x:\limsup_{N}\frac{1}{N}\sum_{1}^{N}f(T^{i}x)>a\right\\}.$

Obviously, $TX_{a}=X_{a}$, therefore, due to the ergodicity of $T$, the
measure of $X_{a}$ is equal to 0 or 1. Let us prove that the first case is
impossible for $a>0$. Let it not be so. Then for (almost) every $x$ there is a
minimal natural number $N(x)$ such that

$\ \frac{1}{N(x)}\sum_{0}^{N(x)-1}f(T^{i}x)>a.$ $None$

The function $N(x)$ is measurable, therefore, recalling real analysis, for any
${\varepsilon}>0$ we find $N$ such that

$\mu(Y_{N})>1-{\varepsilon},\ \ Y_{N}=\\{x:N(x)<N\\}.$

Since we know the Rokhlin-Halmos lemma, it will be convenient for us to use
it. We recall it.

For an ergodic transformation $T$, ${\varepsilon}>0$ and a natural $H$, there
exists a measurable set $B$ such that the sets $T^{i}B$ are disjoint for
$0\leq i\leq H-1$ and

$\mu\left({\bigsqcup}_{i=0}^{H-1}T^{i}B\right)>1-{\varepsilon}.$

Take $H\gg N\gg 1$ and the set $B$ from the lemma ($T^{i}B$ are disjoint for
$0\leq i\leq H$).

Represent the phase space $X$ as the union of the following sets (pieces of
orbits),

$x,Tx,T^{2}x,\dots,T^{h(x)}x,$

where $x$ runs through all $B$, and $h(x)>0$ is chosen so that $T^{h(x)+1}x\in
B$, but the points $Tx,T^{2}x,\dots,T^{h(x)}x$ do not belong to $B$. Note that
such a representation is guaranteed by the Lemma, and $h(x)\geq H$.

Now the most important thing begins. We walk along the set
$x,Tx,\dots,T^{h(x)}x,$ performing the following procedure. If $x\in Y_{N}$,
then $N(x)$ first points, starting with $x$, put into a bag. Look at the next
point, if it does not belong to $Y_{N}$, skip it and look at the next point,
etc. Having encountered a point $x^{\prime}$ from $Y_{N}$, we again put into
the bag
$x^{\prime},Tx^{\prime},T^{2}x^{\prime},\dots,T^{N(x^{\prime})-1}x^{\prime}$
and move on, guided by the described rules. Recall that here
$N(x^{\prime})\leq N\ll H\leq h(x)$. This activity stops when
$x^{\prime}=T^{h}x$ and $H-h<N$. Having performed the procedure for each point
$x\in B$, denoting the bag by $X^{\prime}$, we note that the measure of
$X^{\prime}$ is greater than $1-{\varepsilon}-\frac{N}{H}.$ That is, this
measure can be arbitrarily close to 1.

On all pieces
$x^{\prime},Tx^{\prime},T^{2}x^{\prime},\dots,T^{N(x^{\prime})-1}x^{\prime}$,
the average value of the function $f$ is greater than $a>0$, if you forgot
about this, see $(\ast)$. Therefore, the inequality

$\int_{X^{\prime}}fd\mu>a\mu(X^{\prime})$

holds. Recalling the real analysis, or more precisely the absolute continuity
of the Lebesgue integral, we get a contradiction:

$0=\int_{X}fd\mu>a/2>0.$

$a>0$.

Now let us define for $a>0$

$X_{-a}=\left\\{x:\liminf_{N}\frac{1}{N}\sum_{1}^{N}f(T^{i}x)<-a\right\\}.$

Similarly (or considering $-f$ instead of $f$) we get $\mu(X_{-a})=0$.

We only have to agree with Birkhoff:

$\mu\left(x:\lim_{N}\frac{1}{N}\sum_{1}^{N}f(T^{i}x)=0\right)=1.$

Birkhoff’s theorem significantly used in the proof of the following
interesting statement.

Krygin-Atkinson theorem. Let $f:X\to{\mathbf{Z}}$, $\int_{X}fd\mu=0$, and $T$
be ergodic. Then for any $N$ and almost all $x\in X$ there is $N(x)>N$ such
that

$\sum_{i=0}^{N(x)-1}f(T^{i}x)=0.$

##  4 Properties equivalent to weak mixing

Let us consider the following properties of the transformation $T$.

1\. The transformation $T$ has weak mixing: $T^{n_{i}}\to_{w}\Theta.$

2\. The spectrum of $T$ in the space orthogonal to the constants is continuous
(there are no eigenfunctions).

3\. The product $T\times T$ is ergodic with respect to $\mu\times\mu$.

4\. The transformation $T$ has almost mixing:

$\forall A,B\in{\mathcal{B}}\ \ \frac{1}{N}\sum_{i=1}^{N}|\mu(T^{i}A\cap
B)-\mu(A)\mu(B)|\to 0,\ \ N\to\infty.$

We can say this: most powers of $T$ are close to $\Theta$.

5\. The transformation $T$ has the property: for any ergodic $S$, the product
$S\times T$ is ergodic.

Theorem 4.1. Properties 1-5 are equivalent.

Proof. The implication 1$\to$ 2 is obvious.

Show 2$\to$3 (not 3 implies not 2). Let $T$ be an ergodic transformation and
the product $T\times T$ is not ergodic.

Then there exists a set $D\subset X\times X$ invariant under $T\times T$, and
$0<\mu\times\mu(C)=c<1$. Let $K(x,y)=\chi_{C}(x,y)$. Consider the operator
$P:L_{2}(\mu)\to L_{2}(\mu)$, defined by the formula

$Pf(y)=\int_{X}K(x,y)f(x)d\mu.$

We have

$PConst=P^{\ast}Const=c\,Const.$

Indeed,

$P\,Const(y)=\int_{X}K(x,y)\,Constd\mu=Const\int_{X}K(x,y)d\mu,$

moreover,

$g(y)=\int_{X}K(x,y)d\mu(x)$

is invariant with respect to $T$, therefore, due to the ergodicity of the
transformation $T$, the function $g$ is constant, obviously equal to $c$. For
the operator $P^{\ast}$ we have the same.

Thus, the operators $P,P^{\ast}$ map the space $H$ of functions with zero mean
(it is orthogonal to the constants) to itself. Since $K(x,y)$ is not a
constant, we obtain $P^{\ast}PH\neq\\{0\\}$. (Prove this by showing that
otherwise $P=\,\Theta$.)

The operator $P^{\ast}P$ is a compact self-adjoint operator commuting with the
operator $T$. (The latter follows from the invariance of $K(Tx,Ty)=K(x,y)$.)
By the Hilbert-Schmidt theorem, there exists an eigenfunction $v$:
$P^{\ast}Pv=av$, where the eigenvalue $a$ is nonzero. Since $T$ commutes with
$P^{\ast}P$, all vectors $T^{i}v$ are eigenfunctions of the operator
$P^{\ast}P$ with eigenvalue $a$. Since the operator $P^{\ast}P$ is compact, we
obtain that the space $L$ generated by all vectors $T^{i}v$ is finite-
dimensional. We have $TL=L$, and from the course of linear algebra we know
that the linear operator $T$ on the finite-dimensional complex space $L$ has
an eigenvector. Thus, we have established 2$\to$3.

We prove 3$\to$4\. Let us denote

$c_{i}=\mu(T^{i}A\cap B),\ \ c=\mu(A)\mu(B).$

From the ergodicity of $T$ as $N\to\infty$ we have

$\frac{1}{N}\sum_{i=1}^{N}c_{i}\to c,$

From the ergodicity of $T\times T$ –

$\frac{1}{N}\sum_{i=1}^{N}c_{i}^{2}\to c^{2}.$

(why?). But from what has been said it follows that

$\frac{1}{N}\sum_{i=1}^{N}(c_{i}-c)^{2}\to 0,\ N\to\infty,$

whence, due to the boundedness of $c_{i}$, we have

$\frac{1}{N}\sum_{i=1}^{N}|c_{i}-c|\to 0,\ N\to\infty,$

which is what was required.

The implication 4$\to$1 seems informally obvious. Exercise.

The implication 5$\to$3 is trivial.

4$\to$5\. The idea of the proof is as follows: most of $T^{i}$ are close to
$\Theta$, the operators $\frac{1}{N}\sum_{i=1}^{N}S^{i}$ tend to $\Theta$
(ergodicity of $S$), therefore,

$\frac{1}{N}\sum_{i=1}^{N}S^{i}\otimes T^{i}\to_{w}\Theta\otimes\Theta,$

and this (mixing on average on cylinders) is equivalent to the ergodicity of
$S\times T$ with respect to $\mu\times\mu$.

##  5  On typical properties of transformations

In functional analysis, the Baire theorem on categories plays a significant
role (in the proof of the Banach-Steinhaus theorem and the Banach theorem on
the inverse operator). Baire categories have found applications in ergodic
theory. All transformations form a group $Aut$, on which the complete Halmos
metric is defined:

$\rho(S,T)=\sum_{i=1}^{\infty}2^{-1}\left[\mu(SB_{i}\Delta
TB_{i})+\mu(S^{-1}B_{i}\Delta T^{-1}B_{i})\right],$

where some fixed family of sets $\\{B_{i}\\}$ is dense in ${\mathcal{B}}$. A
family of transformations is typical (massive, generic) if its complement is a
set of the first category, i.e. a countable union of nowhere dense sets. A
property of a transformation is called typical if the set of all
transformations with this property is typical. We will now list the properties
that are important to us. The first two properties are typical. Fix a standard
probability space $(X,{\mathcal{B}},\mu)$ and consider its automorphism group
$Aut$, equipped with the full Halmos metric $\rho$:

$\rho(S,T)=\sum_{i}2^{-i}\left(\mu(SA_{i}\Delta TA_{i})+\mu(S^{-1}A_{i}\Delta
T^{-1}A_{i})\right),$

where the family of sets $\\{A_{i}\\}$ is dense in the algebra
${\mathcal{B}}$. A property is said to be generic if the set of automorphisms
(hereinafter referred to as transformations) with this property contains some
$G_{\delta}$-set dense in $Aut$.

Without proof, we use the fact that the metric is complete and the metric
space $Aut$ is separable (see [1]). If $X$ is a segment $[0,1]$ with Lebesgue
measure, then $Aut$ contains a dense family of all simplest rearrangements of
segments. They are obtained as follows: we split $X$ into segments of equal
length and consider all corresponding segments exchange transformations. We
obtain a countable group of periodic transformations dense in $Aut$.

2.1. Typicalness and non-typicalness of weak limits for powers.

As Rokhlin and Halmos showed, the absence of mixing and the presence of weak
mixing are typical properties. Let us see how to prove more general facts.

A function of the operator $Q(T)$ is called admissible if it has the form

$Q(T)=a\Theta+\sum_{i}a_{i}T^{i},\ \ a,a_{i}\geq 0,\ \sum_{i}a_{i}=1-a,$

where $\Theta$ is the operator of orthoprojection onto the space of constants
in $L_{2}(X,{\mathcal{B}},\mu)$. Let us recall that a transformation $T$
satisfying the condition $T^{m}\to\Theta$ as $m\to\infty$ is called mixing. We
denote the transformation and the transformation-induced operator in
$L_{2}(X,{\mathcal{B}},\mu)$ in the same way.

Theorem 5.1. For an infinite set $M\subset{\mathbf{N}}$ and admissible
functions $Q$,$R$

(i) the set of transformations $T$ such that for some infinite subset
$M(T)\subset M$ $T^{m}\to_{w}R(T)$ holds for $m\in M(T)$ , $m\to\infty$, is
typical;

(ii) the set of transformations $T$ such that $T^{m}\to_{w}Q(T)$ for $m\in
M,m\to\infty$, is a set of the first category.

For $R=I$, $Q=\Theta$, we obtain the above-mentioned results of Halmos and
Rokhlin.

Proof (i). Fix a dense set of transformations $\\{J_{q}\\}$ in $Aut$,
$q\in{\mathbf{N}}$. There exists an infinite subset $M^{\prime}\subset M$ and
a weakly mixing transformation $S$ such that $S^{m}\to_{w}R(S)$, $m\in
M^{\prime}$.

The desired $S$ can be realized as a rank-1 construction (see §10) with
$S^{h_{j}}\to_{w}R(S)$, $h_{j}\in M$.

Let $w$ denote the metric defining the weak operator topology. For any $n$ and
$q$ we find a number $m=m(n,q)\in M$ and a neighborhood $U(n,q)$ of the
transformation $J_{q}^{-1}SJ_{q}$ such that the inequality

$w(T^{m},R(T))<\frac{1}{n}$

is satisfied for all $T\in U(n,q)$. We obtain a $G_{\delta}$-set

$W=\bigcap_{n}\bigcup_{q}U(n,q).$

It is dense in $Aut$, since the conjugacy class of the ergodic transformation
$S$ is dense in $Aut$ (a consequence of the Rokhlin-Halmos lemma).

If $T\in W$, then for any $n$ there exists $q(n)$ such that for
$m(n)=m(n,q(n))\in M$ the inequality $w(T^{m(n)},R(T))<\frac{1}{n}$ holds.
Such $m(n)$ form a set $M(T)\subset M$.

We have thus obtained that $W$ consists of transformations satisfying the
condition of item (i).

Assertion (ii) logically follows from (i). Indeed, fix an infinite $M$, for a
typical transformation $S$ there exists an infinite subset of $M$ on whose
elements $\\{S:S^{m}\to_{w}R(S)\neq Q(S)\\}$. Thus, the condition
$\\{T:T^{m}\to_{w}Q(T)\\}$ for $m\in M,m\to\infty$, is satisfied only for
atypical $T$.

Theorem 5.2. Let $M(m)$ be a fixed sequence such that $M(m)\gg m$. The
following property of transformations $T$ is typical: there is a sequence
$m_{i}\to\infty$ such that any sequence $n_{i}$, $m_{i}\leq n_{i}\leq
M(m_{i})$, is mixing: $T^{n_{i}}\to_{w}\Theta.$

If, for example,

$\LARGE M(m)=m!^{m!^{m!}},$

then we see that sometimes our typical $T$ very … very long time (from $m_{i}$
to some $M>M(m_{i})$) must mix very well. However the typical transformation
after such a mixing epoch for an extremely long time becomes again rigid.

Theorem 5.3. Let a sequence $r_{n}\gg n$ be fixed, the following property of
transformations $T$ is typical: for any $\varepsilon>0$ there is $n$ for which
$\rho(Id,T^{nr})<\varepsilon$ for all $r$, $1\leq r\leq r_{n}$.

In fact there is a more general assertion.

Theorem 5.4. Let $Q$ be admissible and a sequence $N(p)\gg p$ be fixed. The
following property of $T$ is typical: there is an infinite set
$M\subset{\mathbf{N}}$ such that $T^{mp_{m}}\to_{w}Q(T)$ as $m\in M$,
$m\to\infty$, for any sequence $p_{m}$ satisfying $1\leq p_{m}\leq N(p_{m})$.

Exercises. – Prove theorems 5.2, 5.3.

– Show that the weak limits of powers of the transformation form a semigroup.

– If the semigroup $W(T)$ of weak limits of powers of $T$ contains all
operators of the form $\frac{1}{2}(T^{k}+I)$, then the semigroup $W(T)$
contains all possible polynomials of the form $\sum_{i=1}a_{i}T^{i}$,
$a_{i}\geq 0$, $\sum_{i=1}a_{i}=1$.

– If $T^{j^{2024}}\to I$, is it true that $T$ is a periodic transformation?

##  6 Lego to construct transformations

How to construct an ergodic transformation $T$, having a weak limit of powers
of the form $Q(T)=\sum_{i}a_{i}T^{i}$? Below is a description of constructions
allowing to realize as weak limits of transformation powers all admissible
functions $Q$.

Rank one constructions. Let

$\bar{s}_{j}=(s_{j}(1),s_{j}(2),\dots,s_{j}(r_{j}-1),s_{j}(r_{j})),\ \
r_{j}>1,\ s_{j}(i)\geq 0,$

a sequence of integer vectors, and $h_{1}=1$. We will inductively define a
transformation $T$ and its phase space. At each step, what was defined at the
previous ones does not change in the future.

At stage $j$, the transformation $T$ cyclically permutes

$E_{j},TE_{j}T^{2},E_{j},\dots,T^{h_{j}-1}E_{j}.$

Such collection is called a tower. For now $T$ is not defined on
$T^{h_{j}-1}E_{j}.$ We cut the interval $E_{j}$ into $r_{j}$ intervals
$E_{j}^{1},E_{j}^{2},E_{j}^{3},\dots,E_{j}^{r_{j}}$ of the same measure.
Consider the columns

$E_{j}^{i},TE_{j}^{i},T^{2}E_{j}^{i},\dots,T^{h_{j}-1}E_{j}^{i},\
i=1,2,\dots,r_{j}.$

Then we construct $s_{j}(i)$ new intervals above the column with number $i$
and obtain a set of intervals

$E_{j}^{i},TE_{j}^{i},T^{2}E_{j}^{i},\dots,T^{h_{j}+s_{j}(i)-1}E_{j}^{i}$

(the intervals do not intersect and have the same measure). For all $i<r_{j}$,
we set

$T^{h_{j}+s_{j}(i)}E_{j}^{i}=E_{j}^{i+1}.$

Therefore, we have stacked the columns into a tower of stage $j+1$

$E_{j+1},TE_{j+1}T^{2}E_{j+1},\dots,T^{h_{j+1}}E_{j+1},$

where

$E_{j+1}=E^{1}_{j},\ h_{j+1}=h_{j}r_{j}+\sum_{i=1}^{r_{j}}s_{j}(i).$

Continuing the construction, we obtain a measure-preserving transformation $T$
on the union $X$ of all intervals. If the measure $X$ is finite, we normalize
it.

Mixing rank one constructions have the trivial centralizer (D. Ornstein, D.
Rudolph, J. King; and V.V. Ryzhikov, J.-P. Thouvenot for infinite mixing
transformations). S. Kalikow proved that (in the case of probability measure)
mixing rank one constructions possess 2-fold mixing: for any measurable
$A,B,C$

$\mu(A\cap T^{m}B\cap T^{m+n}C)\to\mu(A)\mu(B)\mu(C),\ \ m,n\to+\infty.$

By the way, it is still unknown whether this is true in general — Rokhlin’s
problem on multiple mixing remains open 75 years.

Exercises. – Prove that rank one transformations are ergodic.

–Let $r_{j}=j$, $s_{j}(i)=i$. Prove that $T^{h_{j}}\to\Theta$. Terry Adams
proved that this construction is mixing.

– Are there constructions such that $T^{n_{j}}\to\Theta$, but $T^{2n_{j}}\to
I$?

– Let $r_{j}=2j$, $s_{j}(i)=0$ as $1\leq i\leq j$ and $s_{j}(i)=1$ as $j<i\leq
2j$. This consruction called Katok’s transformation $T$. Prove that it is
weakly mixing, non-mixing, and weak closure of its powers have all operators
in the form $\frac{I}{2}+\frac{T^{k}}{2}$ (in fact all operatos
$P=\sum_{i=1}a_{i}T^{i}$, $a_{i}\geq 0$, $\sum_{i=1}a_{i}=1$).

##  7  Typical entropy invariants

We define the following little modification of the Kirillov-Kushnirenko
entropy. Let $P=\\{P_{j}\\}$ be a sequence of finite subsets in a countable
infinite group $G$. For a measure-preserving action $T=\\{T_{g}\\}$ of $G$, we
define the quantities

$h_{j}(T,\xi)=\frac{1}{|P_{j}|}H\left(\bigvee_{p\in P_{j}}T_{p}\xi\right),$
$h_{P}(T,\xi)={\limsup_{j}}\ h_{j}(T,\xi),$ $h_{P}(T)=\sup_{\xi}h_{P}(T,\xi),$

where $\xi$ denotes a finite measurable partition of $X$, $H(\xi)$ is the
entropy of the partition $\xi$:

$H(\\{C_{1},C_{2},\dots,C_{n}\\})=-\sum_{i=1}^{n}\mu(C_{i})\ln\mu(C_{i}).$

We will be interested only in the case $|P_{j}|\to\infty$ (although the case
of bounded cardinality also makes sense, directly related to the properties of
the type of multiple mixing).

Entropy. If $P_{j}=\\{0,1,2,\dots,j\\}$, then $h_{P}(T)$ is the classical
entropy $h(T)$, see [2].

$P$-entropy for large progressions. Let’s consider the following special case
$G=Z$, when $P_{j}$ are increasing in size progressions:
$P_{j}=\\{j,2j,\dots,L(j)j\\}$, for some sequence $L(j)\to\infty$.

Theorem 7.1. The set $\\{S:h_{P}(S)=\infty\\}$ is typical.

Proof. Let $\\{J_{q}\\}$, $q\in{\mathbf{N}}$, be dense in $Aut$, and let $T$
be the Bernoulli transform, denoted by $T_{q}=J_{q}^{-1}TJ_{q}$. The set
$\\{T_{q}\\}$ is dense in $Aut$. Fix a dense family $\\{\xi_{i}\\}$ of finite
measurable partitions.

For any $n,q$ there exists $j=j(n,q)$ such that for all $i\leq n$ we have

$h_{j}(T_{q},\xi_{i})=\frac{1}{L_{j}}H(\bigvee_{n=1}^{L(j)}T^{nj}_{q}\xi_{i})>H(\xi_{i})-\frac{1}{n}.$
$None$

Indeed, $T_{q}$ is Bernoulli, we find a partition $\xi$ close to a fixed
partition $\xi_{i}$, then for some number $m(i,q,n,)$ the partitions
$T^{nj}_{q}\xi$ are independent for all $n$ and $j>m(i,n,q)$. This implies
$(\ast n\ast)$ for all sufficiently large $j$.

We choose a neighborhood $U(n,q)$ of the transformation $T_{q}$ such that for
all $S\in U(n,q)$ the inequality

$h_{j}(S,\xi_{i})>H(\xi_{i})-\frac{1}{n}.$

The set

$W=\bigcap_{n}\bigcup_{q}U(n,q),$

is dense $G_{\delta}$. If $S\in W$, then for any $n$ there is $q(n)$ such that
the inequality

$h_{j(n,q(n))}(S,\xi_{i})>H(\xi_{i})-\frac{1}{n}$

is satisfied for $i\leq n$. And this leads to $h_{P}(S)=\infty$. Thus the set
$\\{S:h_{P}(S)=\infty\\}$ is typical.

Compact families with zero $P$-entropy. Denote by $E_{0}$ the class of
transformations with zero entropy, $K^{Aut}=\\{J^{-1}SJ:S\in K,J\in Aut\\}$.

Theorem 7.2. If $K\subset E_{0}$ is a compact set and $(Aut,\rho)$, then the
set $K^{Aut}$ is not typical.

Proof. Fix a dense family of finite partitions $\xi_{i}$. If $h(S)=0$, for any
$i$ we have

$h(S^{j},\xi)=\lim_{L\to\infty}\frac{1}{L}H\left(\bigvee_{p=1}^{L}S^{jp}\xi_{i}\right)=0.$

For $S\in K$ and $j$ we find a sequence of progressions $P(S)=\\{P_{j}(S)\\}$

$P_{j}(S)=\\{j,2j,\dots,L_{S}(j)j\\}$

such that

$\frac{1}{|P_{j}(S)|}H\left(\bigvee_{p\in
P_{j}(S)}S^{p}\xi_{i}\right)<\frac{1}{j}$

is satisfied for $i<j$.

Given the structure of the sets $P_{j}(S)$ and the fact that $K$ is compact,
we find a sequence $L(j)\to\infty$ such that for any $S\in K$ and all
sufficiently large $j$ we have $L(j)>L_{S}(j)$. Then for a sequence $P$ of
expanding progressions $P_{j}=\\{j,2j,\dots,L(j)j\\}$ we have $h_{P}(S)=0$ for
all $S\in K$ and thus for all $h_{P}(T)=0$ for all $T\in K^{Aut}$. It follows
from Theorem 7.1 that $K^{Aut}$ is a set of the first category, which
completes the proof.

##  8 Poisson suspensions with completely positive $P$-entropy

Ergodic theory studies large classes of dynamical systems that have an
external origin. These include, in particular, Gaussian and Poisson
automorphisms. The former are associated with the action of the group of all
orthogonal operators on a space with a Gaussian measure, while the latter are
the result of an injective embedding of the group of transformations
preserving the sigma-finite measure into the group of transformations
preserving the Poisson probability measure. For Gaussian automorphisms, see
[2].

The Poisson measure. Consider the configuration space $X_{\circ}$, which
consists of all infinite countable sets $x_{\circ}$ such that each above
interval from the spaces $X$ contains only a finite number of elements of the
set $x_{\circ}$.

The space $X_{\circ}$ is equipped with the Poisson measure. We call its
definition. For a subset $A\subset X$ of a finite $\mu$-measure, we define
configuration subsets $C(A,k)$, $k=0,1,2,\dots$, to $X_{\circ}$ by the formula

$C(A,k)=\\{x_{\circ}\in X_{\circ}\ :\ |x_{\circ}\cap A|=k\\}.$

All possible finite intersections of the form $\cap_{i=1}^{N}C(A_{i},k_{i})$
form a semiring. A Poisson measure $\mu_{\circ}$ is given on this semiring.
Provided that the measurable sets $A_{1},A_{2},\dots,A_{N}$ do not intersect
and have a finite measure, we set

$\mu_{\circ}\left(\bigcap_{i=1}^{N}C(A_{i},k_{i})\right)=\prod_{i=1}^{N}\frac{\mu(A_{i})^{k_{i}}}{k_{i}!}e^{-\mu(A_{i})}.$
$None$

The meaning of this formula is as follows: if the sets $A$, $B$ do not
intersect, then probability $\mu_{\circ}(C(A,k))\cap C(B,m))$ of $k$ points of
configuration $x_{\circ}$ in $A$ simultaneously and $m$ points of the
configuration $x_{\circ}$ in $B$ is equal to the product of the probabilities
$\mu_{\circ}(C(A,k))$ and $\mu_{\circ}(C(B,m))$. In other words, the events
$C(A,k))$ and $C(B,m))$ are independent. Since the sets
$A_{1},A_{2},\dots,A_{N}$ do not intersect, so the product appears in the
formula $(\circ)$. Any element of the semiring is a finite union semiring
elements for which the Poisson measure is defined by $(\circ)$. The measure
extends from the semiring to the Poisson configuration space
$(X_{\circ},\mu_{\circ})$, isomorphic to the standard Lebesgue probability
space.

An automorphism $T$ of the space $(X,\mu)$ naturally induces an automorphism
$T_{\circ}$ of the space $(X_{\circ},\mu_{\circ})$, this $T_{\circ}$ is called
Poisson suspension.

Examples of $T_{\circ}$ with completely positive $P$-entropy. Back to rank one
transformations, let $s_{j}(i)>L(j)h_{j}$, $L(i)\to\infty$. Then
$\mu(X_{j})\to\infty$ and moreover for the correponding rank one construction
$T$ the sets

$X_{j},\ T^{h_{j}}X_{j},\ T^{2h_{j}}X_{j},\ \dots,\ T^{L(j)h_{j}}X_{j}$

do not intersect. The same is automatically true for all $A\subset X_{j}$.

Let $C=C(A,k)$, where $A\subset X_{j_{0}}$, for the Poisson suspension
$T_{\circ}$ we see that the sets

$C,\ T^{h_{j}}_{\circ}C,\ T^{2h_{j}}_{\circ}C\ \dots,\ T^{L(j)h_{j}}_{\circ}C$

are independent with respect to the Poisson measure.

Standard reasoning shows that $T_{\circ}$ has the completely positive
$P$-entropy, where

$P=\\{P_{j}\\},\ \ P_{j}=\\{h_{j},2h_{j},\dots,L(j)h_{j}\\}.$

From the above the following result follows.

Theorem 8.1 (V.V. Ryzhikov, J.-P. Thouvenot). For any ergodic transformation
$S$ of zero entropy, i.e. $h(S)=0$, there is a Poisson suspension $T$ of zero
entropy and a sequence of progression $P$ such that $h_{P}(S)=0$ and
$h_{P}(T)=\infty$.

Remark. In this theorem Poisson suspensions with the same success can be
replaced by Gaussian automorphisms (see [2] for definitions).

##  9 Spectral theorem for unitary operators

Ivertible measure-preserving transformations induce unitary operators on
$L_{2}$. All properties of the latters are completely determined by the so-
called spectral measures. Let us consider two examples.

Let $U:l_{2}({\mathbf{Z}})\to\l_{2}({\mathbf{Z}})$ be a shift in the space
$l_{2}$ of two-sided sequences:

$Ue_{n}=e_{n+1}$

(we can assume that an orthonormal basis $\\{e_{n}\\}$ is given in a Hilbert
space, $U$ is a shift on this system, which naturally extends to a unitary
operator on the entire space).

Consider the operator: $V:L_{2}({\mathbf{T}},\sigma)\to
L_{2}({\mathbf{T}},\sigma)$,

$Vf(z)=zf(z),|z|=1,$

where ${\mathbf{T}}$ is the unit circle in the complex plane, $\sigma$ is the
normalized Lebesgue measure on ${\mathbf{T}}$. Note that the mapping $\Phi$,

$\Phi e_{n}=z^{n},$

(here $z^{n}$ is a function) realizes an isomorphism of the spaces
$\l_{2}({\mathbf{Z}})$ and $L_{2}({\mathbf{T}},\sigma)$ and an isomorphism of
the operators $U$ and $V$. It turns out that $V$ is the spectral
representation of the operator $U$, and $\sigma$ is its spectral measure.

Remark. One of Banach’s open problems is formulated as follows: is there a
transformation with simple Lebesgue spectrum, i.e a transformation that as an
operator is isomorphic to the above operator $U$?

Let us give now a very simple example of a unitary operator in one-dimensional
space: $Ux=-x$ and the multiplication operator $V$ in
$L_{2}({\mathbf{T}},\sigma)$, given by the formula $Vf(z)=zf(z),$ where
$\sigma$ is a measure on ${\mathbf{T}}$ concentrated at a single point $-1$.
This measure is the spectral measure of the operator $U$ (and the operator
$V$).

We now turn to a general case.

Theorem 9.1. If a unitary operator $U:H\to H$ has a cyclic vector, then it is
isomorphic to the operator $V:L_{2}({\mathbf{T}},\sigma)\to
L_{2}({\mathbf{T}},\sigma)$, $Vf(t)=tf(t),|t|=1$, for some Borel measure on
${\mathbf{T}}$.

Proof. Let $h$ be a cyclic vector, i.e. the closure of the space containing
all $U^{n}h,\ n\in{\mathbf{Z}}$, is $H$. Let us define the function $\rho$ on
${\mathbf{T}}$

$\rho_{N}(z)=\frac{1}{N}\left(\sum_{i=0}^{N-1}z^{i}U^{-i}h,\
\sum_{j=0}^{N-1}z^{j}U^{-j}h\right)=\frac{1}{N}\left(\sum_{i,j=0}^{N-1}z^{i-j}U^{j-i}h,\
h\right)\geq 0.$

Since

$\int_{{\mathbf{T}}}z^{i-j}dm=0,\ i\neq j,\ \int_{{\mathbf{T}}}z^{0}dm=1,$

we obtain

$\int_{{\mathbf{T}}}\rho\ dm=\frac{N}{N}=1.$

Consider the sequence of measures $\sigma_{N}$:

$d\sigma_{N}=\rho_{N}(z)dm.$

This sequence, like any sequence of normalized measures on a compact set, has
a weak limit point, which is a normalized Borel measure on the circle
${\mathbf{T}}$:

$\forall f\in C({\mathbf{T}})\ \
\int_{{\mathbf{T}}}fd\sigma_{N_{k}}\to\int_{{\mathbf{T}}}fd\sigma.$

The resulting measure is the desired one.

Since

$\int_{{\mathbf{T}}}z^{i}d\sigma_{N_{k}}=\frac{N_{k}-|i|}{N_{k}}(U^{i}h,h)\to(U^{i}h,h)$

and

$\int_{{\mathbf{T}}}z^{i}d\sigma_{N_{k}}\to\int_{{\mathbf{T}}}z^{i}d\sigma,$

we get

$(U^{i}h,h)=\int_{{\mathbf{T}}}z^{i}d\sigma.$

Thus,

$(U^{i}h,U^{j}h)=\int_{{\mathbf{T}}}z^{i}\bar{z}^{j}d\sigma,$

from which we arrive at the fact that the comparison

$\Phi U^{i}h=z^{i}$

preserves the scalar product and extends to a linear isometry of the spaces
$H$ and $L_{2}(\sigma)$.

The resulting measure $\sigma$ is called the spectral measure, its definition
depended on the choice of a cyclic vector, however, all spectral measures for
the operator $U$ are equivalent to each other.

Now we formulate the spectral theorem for the general case of unitary
operators.

Theorem 9.2. A unitary operator $U:H\to H$ on a separable Hilbert space is
isomorphic to the operator $V$ acting in
$L_{2}({\mathbf{T}}\times{\mathbf{N}},\sigma)$ for some Borel measure $\sigma$
on ${\mathbf{T}}\times{\mathbf{N}}$ by the formula

$Vf(z,n)=zf(z,n),\ \ f\in L_{2}({\mathbf{T}}\times{\mathbf{N}},\sigma).$

The proof is easy to obtain from the previous theorem using the decomposition
of $H$ into an orthogonal sum of cyclic subspaces.

Exercises. – A unitary operator is completely determined (up to isomorphism)
by some measure $\sigma$ on the unit circle ${\mathbf{T}}$ in the complex
plane and the multiplicity function $m(z)$. Let a normal operator be the
multiplication by a function $\phi:X\to{\mathbf{C}}$ on some measure space
$(X,\mu)$, how to find the corresponding spectral measure $\sigma$ on
${\mathbf{C}}$ and the multiplicity function $m(z)$?

Solution. Consider the graph of the function $\phi$ in $X\times{\mathbf{C}}$.
Lift the measure $\mu$ onto it and project $\gamma$ on ${\mathbf{C}}$. We
obtain $\sigma$. The measure $\gamma$ corresponds to a system of conditional
measures $\gamma_{z}$. The function $m(z)$ is the number of points on which
the discrete measure $\gamma_{z}$ is concentrated. If the measure
$\gamma_{z}\neq 0$ is of another kind, we set $m(z)=\infty$.

– Decompose the space on which the unitary operator acts into the orthogonal
sum of its cyclic subspaces.

– Find the spectral measure of the operator induced by rotating the circle by
an angle $a$.

– Prove von Neumann’s theorem by using spectral theorem.

##  10 Compact factors, Kronecker algebra

We begin with some remarks on the eigenfunctions of the ergodic transformation
of $T$.

1\. If $Tf=\lambda f$, then $|f|=const$.
Since $T|f|=|Tf|=|\lambda f|=|f|,$ and $T$ is ergodic, we get $|f|=const$.

2\. If $Tf=\lambda f$ and $Tg=\lambda g\neq 0$, then
$\frac{f}{g}=\frac{Tf}{Tg}=Const$.
In other words, the multiplicity of an eigenvalue of an ergodic transformation
is equal to 1.

3\. The eigenvalues of a measure-preserving transformation form a group.
This follows from the multiplicativity property of operators induced by a
change of variable, for them the equality

$T(fg)=Tf\,Tg.$

holds. Then, if $Tf=\lambda_{1}f$ and $Tg=\lambda_{2}g$, then
$T(fg)=\lambda_{1}\lambda_{2}fg.$

4\. If $Sf=\lambda f\neq Const$ and $Tg=\lambda g\neq Const$, then the product
$S\times T$ is not ergodic.
The function $h(x,y)=\frac{f(x)}{g(y)}=\frac{Sf(x)}{Tg(y)}\neq Const$,
therefore $S\times T$ is not ergodic.

Compact factor coincides with Kronecker algebra. The smallest sigma-algebra
${\mathcal{K}}\subset{\mathcal{B}}$ with respect to which all eigenfunctions
of the transformation $T$ are measurable is called the Kronecker algebra.

The compact factor ${\mathcal{K}}^{\prime}$ of the transformation $T$ is the
algebra of sets generated by compact functions, i.e. functions $f\in L_{2}$
such that the orbit $\\{T^{i}f\\}$ is precompact in $L_{2}$.

Theorem 10.1. The compact factor coincides with the Kronecker algebra:
${\mathcal{K}}={\mathcal{K}}^{\prime}$.

Proof outline. From the spectral theorem we obtain that the space on which the
unitary operator induced by the transformation acts is the orthogonal sum of
two spaces:

the first is generated by the eigenvectors (corresponds to the discrete part
of the spectrum),

and the second is the orthogonal complement to the first. The compact vectors
form exactly the first space.

The space $L_{2}({\mathcal{K}}^{\prime})$ consists of compact vectors, so it
coincides with $L_{2}({\mathcal{K}})$, whence
${\mathcal{K}}={\mathcal{K}}^{\prime}$.

##  11  Progression recurrence for weakly mixing transformations

The famous theorem of Szemeredi (he studied at the Faculty of Mechanics and
Mathematics of Moscow State University) on progressions is equivalent to the
fact that for any set $A$ of positive measure and an invertible measure-
preserving transformation $T$ of the probability space, there is $i>0$ for
which the property of multiple recurrence holds:

$\mu(A\cap T^{i}A\cap T^{2i}A\dots\cap T^{ki}A)\ >0.$

This statement is easy to deduce from the theorem on progressions, but
Furstenberg proved it using the methods of ergodic theory and thus gave a
completely new proof of Szemeredi’s theorem. The case $k=1$ is the Poincare
recurrence theorem. For $k=2$ and especially for $k>2$ the proof is not so
easy as for $k=1$.

Theorem 11.1. For a weakly mixing transformation $T$ of the probability space
$(X,{\mathcal{B}},\mu)$ for any $A,A_{1},\dots A_{k}\in{\mathcal{B}}$ as
$N\to\infty$ we have

$\frac{1}{N}\sum_{i=1}^{N}\mu(A\cap T^{i}A_{1}\cap T^{2i}A_{2}\dots\cap
T^{ki}A_{k})\ \to\ \mu(A)\mu(A_{1})\dots\mu(A_{k}).$ $None$

Let $k=2$. From any infinite subset of the natural numbers we can always
choose a subsequence $N(k)\to\infty$ so that for any
$A,A_{1},A_{2}\in{\mathcal{B}}$

$\frac{1}{N(k)}\sum_{i=1}^{N(k)}\mu(A\cap T^{i}A_{1}\cap T^{2i}A_{2})\ \to\
\nu(A\times A_{1}\times A_{2}),$

where $\nu(A\times A_{1}\times A_{2})$ is for now a notation for the limit of
expressions on the left. It is easy to see that $\nu$ as a function on the
cylinders $A\times A_{1}\times A_{2}$ that form a semiring is a measure. The
invariance is obvious

$\nu(A\times A_{1}\times A_{2})=\nu(TA\times TA_{1}\times TA_{2}),$

and due to averaging there is an additional invariance

$\nu(A\times A_{1}\times A_{2})=\nu(A\times TA_{1}\times T^{2}A_{2}).$

The measure $\nu$ as a measure on a cube is projected onto factors in the
measure $\mu$. Such $T\times T\times T$-invariant measures with good
projections are called self-joinings or joinings.

The weak mixing property is equivalent to the ergodicity of the transformation
$T\times T^{2}$, which in turn means that any sets $A_{1}\times A_{2}$ and
$B_{1}\times B_{2}$ are equidecomposable provided that
$\mu(A_{1})\mu(A_{2})=\mu(B_{1})\mu(B_{2}).$ The latter means that

$A_{1}\times A_{2}\ ={\bigsqcup}_{n=1}^{\infty}(C_{n}\times D_{n}),$
$B_{1}\times B_{2}\ ={\bigsqcup}_{n=1}^{\infty}(T^{p(n)}C_{n}\times
T^{2p(n)}D_{n}),$

where the equalities are satisfied up to sets of zero $\mu\times\mu$-measure.
We get

$\sum_{n}\nu(A\times C_{n}\times D_{n})=\sum_{n}\nu(A\times
T^{p(n)}C_{n}\times T^{p(n)}D_{n}),$ $\nu(A\times A_{1}\times
A_{2})=\nu(A\times B_{1}\times B_{2}).$

From the above it follows that $\nu=\mu\times\mu\times\mu$. And this means

$\frac{1}{N}\sum_{i=1}^{N}\mu(A\cap T^{i}A_{1}\cap T^{2i}A_{2})\ \to\
\mu(A)\mu(A_{1})\mu(A_{2}).$

Exercise. Establish ${\bf(a.p.Mix(k))}$ by induction for all $k>2$.

##  12 Double recurrence for ergodic transformations

Theorem (Roth, Furstenberg). For any set $A$ of positive measure and an
invertible measure-preserving transformation $T$ of the probability space
$(X,{\mathcal{B}},\mu)$ there exists $i>0$ such that

$\mu(A\cap T^{i}A\cap T^{2i}A)\ >0.$

Proof(by J.-P. Thouvenot and V.V. Ryzhikov, last century). Due to Rokhlin’s
theorem on the decomposition of an invariant measure into ergodic components,
the general case reduces to the case of an ergodic transformation $T$, which
we will now consider. The ergodicity of the transformation $T$ is equivalent
to

$\frac{1}{N}\sum_{i=1}^{N}T^{i}f\to_{w}const=\int_{X}f\ d\mu,$

for all $f\in L_{\infty}(\mu)$.

We define the operator $J:L_{2}(\mu)\to L_{2}(\mu\times\mu)$ by the formula

$(Jf,g\otimes h)=\lim_{N_{k}}\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\int_{X}f\
T^{i}g\ T^{2i}h\ d\mu,$

where $f,g,h\in L_{\infty}(\mu)$. The operator $J$ is well defined, since due
to the ergodicity of the transformation $T$ we obtain

$\lim_{N_{k}}\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\int_{X}T^{i}g\ T^{2i}h\
d\mu=\lim_{N_{k}}\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\int_{X}g\ T^{i}h\
d\mu=\int_{X}g\ d\mu\int_{X}h\ d\mu.$

The averaging that appears in the definition of the operator $J$ ensures the
equality

$(T\otimes T^{2})J=J.$

Remark. The equality that has appeared is directly related to the additional
invariance discussed $(Id\times T\times T^{2})\nu=\nu$ for the self-joining
$\nu$ corresponding to the operator $J$:

$\nu(A\times B\times C)=(J\chi_{A},\chi_{B}\otimes\chi_{C}).$

Self-joinings and intertwining operators in some cases are two sides of the
same coin, the choice of sides is dictated by considerations of convenience.
The operator $J$ under consideration is the operator of conditional
mathematical expectation:

$(Jf)(x_{1},x_{2})=\int_{X}f(x)d\nu_{(x_{1},x_{2})},$

where $\nu_{(x_{1},x_{2})}$ is the family of conditional measures
corresponding to the measure $\nu$.

Let us return to the proof. Since $(T\otimes T^{2})Jf=Jf$, the image of the
operator $J$ consists of functions that are fixed with respect to the operator
$T\otimes T^{2}$. But the space of such fixed functions is a subspace of
$V\otimes V$, where $V$ is generated by all eigenvectors of the operator $T$.
This follows from the standard facts of the spectral theory of unitary
operators, prove it using the spectral theorem.

Denote by $\pi$ the orthoprojection of $L_{2}(\mu)$ onto $V$.

Exercise. The operator $\pi$ preserves the non-negativity of the function.

We have

$(T\otimes T^{2})Jf=Jf,\ \ (\pi\otimes\pi)J=J,$ $(Jf,g\otimes
h)=(Jf,(\pi\otimes\pi)g\otimes h)=$
$\lim_{N_{k}}\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\int_{X}f\ T^{i}\pi g\ T^{2i}\pi
h\ d\mu=\lim_{N_{k}}\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\int_{X}\pi f\ R^{i}\pi
g\ R^{2i}\pi h\ d\mu,$

where $R$ is the constraint of $T$ on $V$.

Exercise. We replaced $f$ with $\pi f$, why?

But the powers of the operator $R$ regularly turn out to be close to the
identity operator, in other words, (again the problem) the set of those $i$,
when

$\|R^{i}\pi f-\pi f\|<{\varepsilon},\ \|R^{2i}\pi f-\pi f\|<{\varepsilon}$

has positive density.

Setting $f=g=h=\chi_{A}$ subject to $\mu(A)>0$, we obtain

$(J\chi_{A},\chi_{A}\otimes\chi_{A})=\lim_{N_{k}}\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\int_{X}\pi\chi_{A}\
R^{i}\pi\chi_{A}\ R^{2i}\pi\chi_{A}\ d\mu\ >\ 0.$

We have thus shown that the inequality $\mu(A\cap T^{i}A\cap T^{2i}A)\ >0$
holds for an infinite set of values of $i$, which completes the proof.

## References

  * [1] Halmos P.R., Lectures on Ergodic Theory. Publ. Math.Soc. Japan, No. 3, Tokyo, 1956
  * [2] Kornfeld I.P., Sinai Ya.G., Fomin S.V., Ergodic Theory, Moscow, 1980