# Scaling entropy and automorphisms with purely point spectrum111Partially
supported by the grants RFBR-08-01-00379-a and RFBR-09-01-12175-ofi-m.

A. M. Vershik

###### Abstract

We study the dynamics of metrics generated by measure-preserving
transformations. We consider sequences of average metrics and
$\epsilon$-entropies of the measure with respect to these metrics. The main
result, which gives a criterion for checking that the spectrum of a
transformation if purely point, is that the scaling sequence for the
$\epsilon$-entropies with respect to the averages of an admissible metric is
bounded if and only if the automorphism has a purely point spectrum. This
paper is one of a series of papers by the author devoted to the asymptotic
theory of sequences of metric measure spaces and its applications to ergodic
theory.

To the memory of my friend Misha Birman

###### Contents

  1. 1 Introduction
  2. 2 Admissible metrics
  3. 3 Average and maximal metrics
  4. 4 Entropy and scaling entropy
    1. 4.1 The $\varepsilon$-entropy of a measure in a metric space
    2. 4.2 Scaling sequence and scaling entropy
  5. 5 Invariant metrics on groups and averages of admissible metrics
    1. 5.1 Invariant metrics and discrete spectrum
    2. 5.2 Admissible invariant metrics
    3. 5.3 Admissibility of the average metric
  6. 6 Criterion for the discreteness and continuity of the spectrum in terms of the scaling entropy
  7. 7 Comparison with the traditional approach, the Pascal automorphism, and concluding remarks
    1. 7.1 Supremum metrics
    2. 7.2 Application of the discreteness criterion
    3. 7.3 The dynamics of metrics

##  1 Introduction

Among the many mathematical and nonmathematical problems we have been
discussing with Misha Birman for many years after our acquaintance began in
the early 1960s, the most intriguing one was the parallel between scattering
theory and ergodic theory. Recently, I have returned to the (yet nonexistent)
“ergodic scattering theory” and some forgotten questions related to it.
However, this paper deals with another subject, which also correlates with M.
Sh. Birman’s research.

It is well known that the problem of deciding whether or not the spectrum of a
given (say, differential) operator is discrete is quite difficult. Several
remarkable early papers by M. Sh. Birman (in the first place, [1]) dealt with
exactly this problem. In ergodic theory and the theory of dynamical systems,
this problem (whether or not the system of eigenfunctions is complete) is also
very difficult. In what follows, we use the terminology common in ergodic
theory that slightly differs from the one adopted in operator theory: we use
the term “discrete spectrum” as a synonym for “pure point spectrum.” This is
justified by the fact that a discrete spectrum in the sense of operator theory
almost never appears in the theory of dynamical systems, since almost always
one deals with unitary operators.

As an example demonstrating the difficulty of this problem, we can mention the
theory of substitutions, or stationary adic transformations [2], on which an
extensive literature exists. The most intriguing problems concern
nonstationary adic transformations with subexponential growth of the number of
vertices in the corresponding Bratteli–Vershik diagram (see [2]). The simplest
and most popular example of such a transformation is the Pascal automorphism
defined in [3]; in this case, the measure space (i. e., the phase space of the
dynamical system) is the space of infinite paths in the Pascal graph endowed
with a Bernoulli measure, and the transformation sends a path to its successor
in the natural lexicographic order. In spite of many efforts, we still do not
know whether the corresponding unitary operator has a discrete (i. e., pure
point) or mixed spectrum, or, which seems most likely, its spectrum is purely
continuous. Attempts to directly construct its eigenfunctions failed; another
approach, based on the characterization of systems with discrete spectrum in
terms of Kushnirenko’s sequential entropy, [10] has not been carried out.

In the paper [14] (for more details, see [4, 8, 27]), we suggested a new
notion, the so-called _scaling entropy_ , which generalizes the notion of
Kolmogorov’s entropy. The main point is that we suggest to average the shifts
of the metric with respect to a given transformation and then compute the
$\varepsilon$-entropy of the average metric. The class of increasing sequences
of positive integers that normalize the growth of this $\varepsilon$-entropy
over all admissible metrics does not depend on the choice of an admissible
metric, so that the asymptotics of the growth of these sequences is a new
metric invariant of automorphisms. Admissible metrics are measurable metrics
satisfying some conditions that do or do not depend on the automorphism (see
Section 2). It is important that we consider not merely the
$\varepsilon$-entropy of a metric, but the $\varepsilon$-entropy of a metric
in a measure space. Admissible metrics play the same role as measurable
generating partitions in the classical theory of Kolmogorov’s entropy
according to Sinai’s definition. At first sight, the difference between using
partitions and metrics looks rather technical: a partition determines a
semimetric of a special form (a so-called “cut semimetric,” see [18]).
However, our approach has two important differences from the classical theory.
First, we use the $\varepsilon$-entropy of the iterated metric on a measure
space rather than the entropy of a partition; this is a generalization of
Kolmogorov’s entropy, which allows one to distinguish automorphisms with zero
entropy. The second, more important, difference is that we use the average
metric (rather than the supremum of metrics, which corresponds to the supremum
of partitions), which has no interpretation in terms of partitions and which
contains more information about the automorphism than the supremum of metrics.

In this paper, we give a necessary and sufficient condition for an
automorphism to have a discrete spectrum in terms of the scaling sequence. The
condition is that this sequence is bounded. This result generalizes a theorem
of S. Ferenczi [15, 16], who considered the measure-theoretic complexity of
symbolic systems by analogy with the ordinary complexity in symbolic dynamics.
Our approach substantially differs from that of [15]: we consider an arbitrary
admissible metric rather than the Hamming metric only, and, which is most
important, introduce the average metric and show that it is admissible in many
cases and, in particular, for the Hamming metric.

Thus criterion, i.e., the boundedness of the growth of the scaling entropy,
should first be applied to adic transformations, e.g., to the Pascal
automorphism (see [4]). Although it is not yet applied to checking that the
spectrum of the Pascal automorphism is not discrete, 222See the footnote in
the subsection 7.2, the corresponding combinatorics is already developed and
described in the recent paper [19], where a lower bound is obtained on the
scaling sequence of the sup metric for the same Pascal automorphism.
Supposedly, one can extend this bound ($\ln n$) to the average metric using
the same techniques.

A wider context is presented in our papers [4, 8], where we suggest a plan for
the study of the dynamics of metrics in a measure space as a source of new
invariants of automorphisms. It is important that the notion of scaling
entropy provides an answer to the question of whether or not the spectrum of a
given transformation is discrete.

Sections 2–4 are of general nature and are intended not only for the purposes
of this paper, which is devoted mostly to automorphisms with discrete
spectrum. Here we introduce our main objects: admissible metrics, averages,
scaling sequences, and the scaling entropy of an automorphism. In Section 5,
we study the dynamics of metrics on a group and find conditions under which
the average metric is admissible. The main result is given in Section 6, where
we present a criterion for checking whether the spectrum of an automorphism is
completely or partially discrete. In the last section, we sketch possible
applications, links to the ordinary construction of entropy, and general
remarks about the dynamics of metrics.

##  2 Admissible metrics

We consider various metrics and semimetrics on a measure space
$(X,{\mathfrak{A}},\mu)$. In what follows, it is assumed to be a standard
space with continuous measure $\mu$ and $\sigma$-algebra $\mathfrak{A}$ of
$\bmod 0$ classes of measurable sets, i.e., a Lebesgue space with continuous
measure in the sense of Rokhlin (see [24]). The space $X\times X$ is endowed
with the $\sigma$-algebra $\mathfrak{A}\times\mathfrak{A}$ and the measure
$\mu\times\mu$.

We define a class of (semi)metrics on a measure space, which plays an
important role in what follows.

######  Definition 1.

A measurable function $\rho:X\times X\rightarrow{\mathbb{R}}_{+}$ is called an
admissible (semi)metric if

1) $\rho$ is a (semi)metric in the ordinary sense on a subset
$X^{\prime}\subset X$ of full measure ($\mu X^{\prime}=1$), i.e.,
$\rho(x,y)\geq 0$, $\rho(x,y)=\rho(y,x)$, $\rho(x,y)+\rho(y,z)\geq\rho(x,z)$
for all triples $(x,y,z)\in X^{\prime}\times X^{\prime}\times X^{\prime}$,
$(\mu\times\mu)\\{(x,x),x\in X^{\prime}\\}=0$, and

$\int_{X}\int_{X}\rho(x,y)d\mu(x)d\mu(y)<\infty.$

In order to formulate the next condition, observe that if $\rho$ satisfies
Condition 1), then the partition $\psi_{\rho}$ of the space $X$ into the
classes of points $C_{x}=\\{y:\rho(x,y)=0\\}$, $x\in X$, is measurable. Hence
we have a well-defined quotient space $X_{\rho}\equiv X/{\psi_{\rho}}$ endowed
with the quotient metric denoted by the same letter $\rho$ and the quotient
measure $\mu_{\psi_{\rho}}\equiv\mu_{\rho}$. For the quotient space
$(X_{\rho},\rho,\mu_{\rho})$, Condition 1) is still satisfied.

2) The (completion of the) metric space $(X,\rho)$ (if $\rho$ is a metric) or
$(X_{\rho},\rho)$ (if $\rho$ on $X$ is a semimetric) is a Polish (= metric,
separable, complete) space with a Borel probability measure $\mu$
(respectively, $\mu_{\rho}$).

Following the measure-theoretic tradition, we must identify (semi)metrics (and
hence the corresponding spaces) if they coincide almost everywhere as
measurable functions on the space $(X\times X,\mu\times\mu)$. Of course, a
(semi)metric that coincides almost everywhere with an admissible (semi)metric
is admissible.333We can define the notion of almost metric as a measurable
function for which all axioms on metric satisfy for almost all pairs or
triples points. As F.Petrov noticed for each almost metric there exists the
admissible metric in our sense which almost everywhere coincided with it.

Condition 2) means that the $\sigma$-algebra of Borel sets in the metric space
$(X,\rho)$ (or $(X_{\rho},\rho)$ in the case of a semimetric) is dense in the
$\sigma$-algebra $\mathfrak{A}$ of all measurable sets and, therefore, the
measure $\mu$ (respectively, $\mu_{\rho}$) is a Borel probability measure. It
is obvious from the definition that a semimetric $\rho$ is admissible if and
only if the corresponding metric in the quotient space $X_{\rho}$ is
admissible.

An equivalent definition of an admissible metric is as follows: almost every
pair of points can be separated by balls of positive measure, or, in other
words, the $\sigma$-subalgebra generated by the open balls $\bmod 0$ separates
points of the space.

One can also formulate the admissibility condition in terms of the notion of a
pure function from our paper [6]:

######  Lemma 1.

A metric $\rho$ is admissible if and only if it satisfies Condition 1) and,
regarded as a function of two variables, is pure in the sense of [6]; the
latter means that the partition of the space on the classes of equivalence
$x\sim y\Leftrightarrow\mu\\{z:\rho(x,z)=\rho(y,z)\\}=1$ is the partition on
the separate points $mod0$. In other words: the map $x\mapsto\rho(x,.)$ is
injective from measure space $(X,\mu)$ to the classes of $mod0$ equal
functions of one variable. 444In other words, almost every point is uniquely
determined by the collection of distances from this point to all points of
some set of full measure (which may depend on the point).

If $\rho$ is a semimetric, then this condition must hold for the metric $\rho$
on the quotient space $(X_{\rho},\rho,\mu_{\rho})$.

Indeed, the purity condition implies that the $\sigma$-algebra of sets
generated by the balls separates points and hence is dense in the
$\sigma$-algebra $(X,{\mathfrak{A}},\mu)$; this also implies the separability.
The converse immediately follows from the definition of an admissible metric.

It is well known (see [24]) that if $(X,\rho)$ is a Polish space, then every
nondegenerate Borel probability measure $\mu$ on $X$ turns $(X,\rho)$ into a
Lebesgue space. In other words, the metric $\rho$ on a Polish space $(X,\rho)$
endowed with a Borel probability measure $\mu$ is an admissible metric on the
space $(X,\mu)$.

As in other our papers, in the definition of an admissible metric we reverse
the tradition and consider various metrics and semimetrics on a fixed measure
space rather than various measures on a given metric space. Recall that
triples $(X,\rho,\mu)$, consisting of a metric space endowed with a measure,
in M. Gromov’s book [5] were called $mm$-spaces, and in the paper [7], metric
triples or Gromov triples.

It is useful to regard admissible metrics as densities of some finite measures
equivalent to the measure $\mu\times\mu$ on $X\times X$:

$dM_{\mu,\rho}=\rho(x,y)d\mu(x)d\mu(y).$

If we set $\int_{X\times X}\rho(x,y)d\mu(x)d\mu(y)=1$, which can be done by
normalizing the metric, then the new measure is also a probability measure.
Observe the following important property of admissible metrics implied by this
interpretation.

######  Theorem 1.

For almost every point $x\in X$, there is a uniquely defined $\bmod 0$
conditional measure $\mu^{x}$ on $X$, which is given by the formula
$d\mu^{x}(A)=\int_{A}\rho(x,y)d\mu(y)$ for every measurable set $A\subset X$.
The family of measures $\\{\mu^{x};x\in X\\}$ satisfies the condition

$\mu(A)=\int_{X}\mu^{x}(A)d\mu(x).$

The metric $\rho$, regarded as a metric on the measure space $(X,\mu^{x})$, is
admissible.

###### Proof.

Consider the new measure $M_{\mu,\rho}$ on the space $X\times X$ and the
measurable partition into the classes of points $C^{x}=\\{(x,*)\\}\subset
X\times X$, and use the classical theorem on the existence of conditional
measures (see, e.g., [24]), which implies the desired formula and the
uniqueness $\bmod 0$ of the family of conditional measures. Now consider the
space $(X,\rho,\mu^{x})$ for a fixed $x$; the metric $\rho$ on this space is
admissible since the admissibility of a metric is obviously preserved under
replacing a measure with an equivalent one. ∎

The conditional measure $\mu^{x}(A)$ can be interpreted as the “average
distance,” or the conditional expectation of the distance from the set $A$ to
the point $x$.

In these terms, the condition that the metric, regarded as a function of two
variables, is pure means that the conditional expectations do not coincide
$\mod 0$ for almost all pairs of points.

Now we can give a convenient criterion of admissibility.

######  Proposition 1.

A measurable function $\rho(\cdot,\cdot)$ satisfying Condition 1) of
Definition 1 is admissible if and only if the following non-degeneracy
condition holds:

for $\mu$\- almost all $x$, the measure of arbitrary balls of positive radius
is positive: $\forall\varepsilon>0,\quad\mu\\{y:\rho(x,y)<\epsilon\\}>0$,
which is equivalent to:

$\mu^{x}([0,\epsilon])>0$

for almost all $x$ and all positive $\epsilon$.

###### Proof.

The fact that an admissible metric satisfies the condition in question was
observed above. Now assume that this condition is satisfied. We must prove
that the space $(X,\rho)$ (if $\rho$ is a metric) or the quotient space
$X_{\rho}$ (if $\rho$ is a semimetric) is separable. It suffices to consider
the case of a metric.

The condition stated in the proposition implies that for every $\epsilon$
there exists $\delta=\delta(\epsilon)$ with $\lim_{\epsilon\to
0}\delta(\epsilon)=0$ such that some set of measure $>1-\delta$ contains a
finite $\epsilon$-net. But this means exactly that the space is separable and
the measure is concentrated on a $\sigma$-compact set. ∎

Several examples. 1\. An important example is the following class of
semimetrics, which was in fact intensively used in entropy theory. Every
partition $\xi$ of a space $(X,\mu)$ into finitely or countably many
measurable sets gives rise to a semimetric:

$\rho_{\xi}(x,y)=\delta_{(\xi(x),\xi(y))},$

where $\xi(z)$ is the element of $\xi$ that contains $z$. In this case,
$X_{\rho}$ is a finite or countable metric space. Such (finite) semimetrics
are called cuts, and their linear combinations are called cut semimetrics (in
the terminology of [18]). It is easy to see that cut (semi)metrics are
admissible.

2\. The very important metric defined by the formula $\rho(x,y)={\rm const}$
for $x\neq y$ determines a discrete uncountable space. We will call it the
constant metric. The constant metric on a space with continuous measure is not
admissible. In this case, the $\sigma$-algebra generated by the open sets is
trivial and does not separate points of the space, i.e., a point is not
determined by the collection of distances to the other points.

3\. The condition defining an admissible metric can be strengthened by
requiring, in condition 1), that

$\int_{X}\int_{X}\rho(x,y)^{p}d\mu(x)d\mu(y)<\infty$

for $p>1$; in this case, we say that the metric is $p$-admissible. Let us say
that $\infty$-admissible metrics (semimetrics) are bounded; this class of
admissible metrics will be most useful in what follows.

4\. In combinatorial examples, it often suffices to consider metrics with
which the space is compact or precompact (or, in the case of a semimetric,
quasi-compact). For example, adic transformations [2, 3] act in the space of
infinite paths of an $\mathbb{N}$-graded graph, which is a totally
disconnected compact space.

The set of admissible metrics on a Lebesgue space $(X,\mu)$ with continuous
measure is a convex cone ${\cal R}(X,\mu)={\cal R}$ with respect to the
operation of taking a linear combination of metrics with nonnegative
coefficients. This cone is a canonical object (by the uniqueness of a Lebesgue
space up to isomorphism) and plays a role similar to the role of the simplex
of Borel probability measures in topological dynamics. In many cases, it
suffices to consider admissible (semi)metrics that produce compact spaces
(after completion), but we do not exclude the case of a noncompact space. One
may consider different topologies on the cone ${\cal R}$; the most natural of
them is the weak topology, in which an $\varepsilon$-neighborhood of a metric
$\rho$ is the collection of metrics

$\Bigl{\\{}\theta:(\mu\times\mu)\\{(x,y):|\rho(x,y)-\theta(x,y)|<\varepsilon\\}>1-\varepsilon\Bigr{\\}}.$

The property of being an admissible metric is invariant under measure-
preserving transformations: if a metric $\rho$ is admissible and a
transformation $T$ of the space $(X,\mu)$ preserves the measure $\mu$, then
the image $\rho^{T}$ of $\rho$, defined by the formula
$\rho^{T}(\cdot,\cdot)=\rho(T\cdot,T\cdot)$, is also admissible. Thus the
group of (classes of) measure-preserving transformations acts on the cone
${\cal R}$ in a natural way.

##  3 Average and maximal metrics

Let $T$ be a measure-preserving transformation (in what follows, it will be an
automorphism). When considering automorphisms or groups of automorphisms in
spaces with admissible semimetrics, it is natural to assume that there exist
an invariant set of full measure on which the (semi)metric is admissible in
the sense of our definition.

In addition to the notion of admissible (semi)metrics, we define the class of
$T$-admissible metrics. The $T$-admissibility condition must be invariant, in
the sense that if $\cal M$ is the class of $T$-admissible metrics, then

$V{\cal M}\equiv\\{\rho:\rho(x,y)=\rho_{1}(Vx,Vy),\;\rho_{1}\in\cal M\\}$

is the class of $VTV^{-1}$-admissible metrics in the same sense. Among many
possible versions, we choose the class of admissible (semi)metrics for which
$T$ is a Lipschitz transformation almost everywhere: there exists a positive
constant $C$ such that for $(\mu\times\mu)$-almost all pairs $(x,y)$, the
condition

$\rho(Tx,Ty)\leq C\rho(x,y)$

holds; let us say that metrics from this class are Lipschitz $T$-admissible
(semi)metrics.

The choice of an appropriate class depends on the problem under consideration
and the properties of the automorphism. For example, in the case of adic
automorphisms, it is most convenient to consider the class of Lipschitz
metrics. This class can also be defined for countable groups of automorphisms.

For an arbitrary admissible semimetric $\rho$, we have defined the partition
$\psi_{\rho}$ of the space $(X,\mu)$. In the same way, given an arbitrary
automorphism $T$ of the space $(X,\mu)$, we consider the $T$-invariant
partition $\psi_{\rho}^{T}=\bigvee_{k=0}^{\infty}T^{k}\psi_{\rho}$. We say
that the semimetric $\rho$ is generating for $T$ if $\psi_{\rho}^{T}$ is the
partition into separate points $\bmod 0$ (which we denote by $\varepsilon$).
If $\rho$ is the metric generated by a finite partition, then this is the
ordinary definition of a generator (see [25]).

Let us define the average metric and the $\sup$-metric for a given
automorphism.

######  Definition 2.

Let $T$ be an automorphism of a space $(X,\mu)$, and let $\rho$ be an
admissible metric.

The average metric $\rho_{n}^{T}$ is defined by the formula

$\hat{\rho}_{n}^{T}(x,y)=\frac{1}{n}\sum_{k=0}^{n-1}\rho(T^{k}x,T^{k}y).$

The $\sup$-metric is defined by the formula

${\bar{\rho}}_{n}^{T}(x,y)=\sup_{0\leq k<n}\rho(T^{k}x,T^{k}y).$

The following important result is a direct corollary of the pointwise ergodic
theorem.

######  Theorem 2.

For any automorphism $T$ and any admissible (semi)metric $\rho$, the limit of
the sequence of average (semi)metrics $\rho^{T}_{n}$, which we denote by
$\hat{\rho}$, exists almost everywhere in the space $(X\times
X,\mu\times\mu)$:

$\hat{\rho}^{T}(x,y)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\rho(T^{k}x,T^{k}y);$

$\hat{\rho}$ is a metric if and only if $\rho$ is a metric or a generating
semimetric.

The existence of the limit follows from the fact that the integral
$\int_{X}\int_{X}\rho(x,y)d\mu(x)d\mu(y)$ is finite and the ergodic theorem.

######  Definition 3.

The metric

$\hat{\rho}^{T}(x,y)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\rho(T^{k}x,T^{k}y)$

(the limit exists $\mu$-a.e.) is called the average, or the $l^{1}$-average of
$\rho$ with respect to the automorphism $T$.

The metric

$\bar{\rho}^{T}(x,y)=\sup_{k\geq 0}\rho(T^{k}x,T^{k}x)$

is called the limiting $\sup$-metric of $\rho$ with respect to $T$.

It is clear that $\hat{\rho}^{T}$ and $\bar{\rho}^{T}$ satisfy all conditions
of the definition of a (semi)metric; in what follows, we will consider only
admissible metrics or generating semimetrics $\rho$, so that $\hat{\rho}^{T}$,
and $\bar{\rho}^{T}$ for an ergodic automorphism $T$, are metrics. The
superscript $T$ in the notation for the average and $\sup$-metrics will be
omitted if the automorphism is clear from the context. It is also obvious that
$\hat{\rho}\leq\bar{\rho}$. However, the metrics $\hat{\rho},\bar{\rho}$ may
not be admissible even if the (semi)metric $\rho$ is admissible.

In what follows, we will mainly consider the average metric $\hat{\rho}$. In
most interesting cases, namely if $T$ is weakly mixing, i.e., its spectrum in
the orthogonal complement to the space of constants is continuous, this metric
is constant and hence not admissible; the $\sup$-metric may be constant even
for automorphisms with discrete spectrum. However, in all cases we will be
interested not in the limiting metrics themselves, but in the asymptotic
behavior of the averages $\rho_{n}^{T}$ as $n\to\infty$. As we will see,
automorphisms with discrete spectrum stand apart, since for them the average
metric is often admissible.

It is clear that $\hat{\rho}$ is nothing else but the projection of the
function $\rho$, regarded as an element of the space $L^{1}(X\times
X,\mu\times\mu)$, to the subspace of $(T\times T)$-invariant functions, i.e.,
the expectation of the metric $\rho$ with respect to the subspace of invariant
functions on $X\times X$. This space consists of constants if and only if $T$
has no nontrivial eigenfunctions (in other words, $T$ is weakly mixing). In
this case, $\hat{\rho}$ is almost everywhere a constant, which is equal to the
average $\rho$-distance between the points of the space $X$. At the same time,
if $T$ is not weakly mixing, then the spectrum of $T$ contains a discrete
component and $\hat{\rho}$ may be a nonconstant $T$-invariant (semi)metric. In
this case, one may obtain bounds on $\hat{\rho}$ using Fourier analysis.

Definitions and Lemma 1 imply the following lemma.

######  Lemma 2.

Let $\rho$ be an admissible Lipschitz metric for an automorphism $T$; then the
average metric $\hat{\rho}$ is also admissible and Lipschitz for $T$.

We do not prove this assertion, because we do not need it in this paper.

Here is an example of computing the average metric generated by the cut
semimetric in the case of a rotation of the circle.

Example. Let $X={\mathbb{T}}^{1}={\mathbb{R}}/\mathbb{Z}$, and let $\lambda\in
X$ be an irrational number. Consider the semimetric
$\rho(x,y)=|\chi_{A}(x)-\chi_{A}(y)|$, where $\chi_{A}$ is the indicator of a
measurable set $A\subset X$; the metric $\rho(x,y)$ is $T$-admissible in the
sense of our definition for the shift $T_{\lambda}$ by any irrational number
$\lambda$. The corresponding average metric is shift-invariant and looks as
follows: $\hat{\rho}(x,y)=m[(A+x)\Delta\mu(A+y)]=m[A\Delta(A+x-y)]$; it is
obviously admissible.

##  4 Entropy and scaling entropy

###  4.1 The $\varepsilon$-entropy of a measure in a metric space

Recall that the $\varepsilon$-entropy of a compact metric space $(X,\rho)$ is
the function $\varepsilon\mapsto H_{\rho}(\varepsilon)$ whose value is equal
to the minimum number of points in an $\varepsilon$-net of $(X,\rho)$.

######  Definition 4.

The $\varepsilon$-entropy of a measure space $(X,\mu)$ with an admissible
metric $\rho$ is the function

$\varepsilon\mapsto
H_{\rho,\varepsilon}(\mu)=\inf\\{H(\nu):k_{\rho}(\nu,\mu)<\varepsilon\\},$

where $\nu$ ranges over the set of all discrete measures with finite entropy
and $k_{\rho}(\cdot,\cdot)$ is the Kantorovich metric on the space of Borel
probability measures on $(X,\rho)$. The entropy of a discrete measure
$\nu=\sum_{i}c_{i}\delta_{x_{i}}$ is defined in the usual way:
$H(\nu)=-\sum_{i}c_{i}\ln c_{i}$.

For our purposes, it is more convenient to use in the above definition another
characteristic instead of $H(\mu)$, namely,

$H^{\prime}_{\rho,\varepsilon}(\mu)=\min\Bigl{\\{}\ln k:\exists
X^{\prime},\mu(X^{\prime})>1-\varepsilon,\;\exists\\{x_{i}\\}_{1}^{k}:X^{\prime}\subset\bigcup_{i=1}^{k}V_{\varepsilon}(x_{i})\Bigr{\\}},$

where $V_{\varepsilon}(x)$ is the $\varepsilon$-ball centered at $x$; thus
$\\{x_{1},x_{2},\dots,x_{k}\\}$ is an $\varepsilon$-net in $X^{\prime}$. The
finiteness of $H^{\prime}$ follows from the fact that a Borel probability
measure in a Polish space is concentrated, up to $\varepsilon$, on a compact
set; $H^{\prime}$ is more convenient for computations than $H$.

In this paper, we use the following simple inequality.

######  Lemma 3.

For every compact metric space $(X,\rho)$ and every nondegenerate Borel
measure $\mu$, the following inequality holds:

$H_{\rho,(d+1)\varepsilon}(\mu)\leq H^{\prime}_{\rho,\varepsilon}(\mu),$

where $d$ is the diameter of the space.

###### Proof.

Assume that the diameter of the compact space does not exceed $1$. Assume that
the measure of a set $X^{\prime}$ is greater than $1-\varepsilon$ and
$X^{\prime}\subset\bigcup_{i=1}^{k}V_{\varepsilon}(x_{i})$. Thus the points
$x_{1},\dots,x_{k}$ form an $\varepsilon$-net in $X^{\prime}$. Consider the
discrete measure $\nu$ supported by the points $x_{1},\dots,x_{k}$ with
charges $\nu(x_{i})$ equal to the measures $\mu(V(x_{i}))$ of the
corresponding balls (if two balls have a nonempty intersection, then we
distribute the measure of the intersection proportionally between their
centers). Choose an arbitrary point $x_{\infty}$ and set its charge equal to
$1-\mu(X^{\prime})$. Now consider the Monge–Kantorovich transportation problem
with input measure $\mu$ and output measure $\nu$. It is easy to see that we
have in fact determined an admissible plan $\Psi$ for this problem: the
transportation from a point $x\in X^{\prime}$ goes to the points $x_{i}$ for
which $x\in V_{\varepsilon}(x_{i})$, and the remaining part of the measure
$\mu$ on the set $X\setminus X^{\prime}$ goes to the point $x_{\infty}$. It is
easy to compute the cost of this plan; this gives a bound on the Kantorovich
distance between the measures $\nu$ and $\mu$:

$k_{\rho}(\nu,\mu)\leq\varepsilon(1-\varepsilon)+\varepsilon<2\varepsilon.$

On the other hand, we have $H(\nu)\leq\ln
k=H^{\prime}_{\rho,\varepsilon}(\mu)$. ∎

###  4.2 Scaling sequence and scaling entropy

Let us define the notion of a scaling sequence for the entropy of an
automorphism. If an automorphism $T$ is fixed, we omit the superscript in the
notation for the average entropy and white simply $\hat{\rho}_{n}$.

######  Definition 5.

Let $T$ be an automorphism of a Lebesgue space $(X,\mu)$ with a $T$-invariant
measure $\mu$. By definition, the class of scaling sequences for the
automorphism $T$ and a given (semi)metric $\rho$ on $X$ is the class, denoted
by ${\cal H}_{\rho,\varepsilon}(T)$, of increasing sequences of positive
numbers $\\{c_{n},n\in{\mathbb{N}}\\}$ such that

${\cal
H}_{\rho,\varepsilon}(T)=\Bigl{\\{}\\{c_{n}\\}:0<\liminf_{n\to\infty}\frac{H_{\hat{\rho}_{n},\varepsilon}(\mu)}{c_{n}}\leq\limsup_{n\to\infty}\frac{H_{\hat{\rho}_{n},\varepsilon}(\mu)}{c_{n}}<\infty\Bigr{\\}}.$

In many cases, the class of scaling sequences for a given metric $\rho$ does
not depend on sufficiently small $\varepsilon$. In this case, it is obvious
that all sequences $\\{c_{n}\\}$ from ${\cal H}_{\rho}(T)$ are equivalent.

######  Definition 6.

Assume that for a given ergodic automorphism $T$ of a space $(X,\mu)$ there
exists a (semi)metric $\rho_{0}$ such that the class of scaling sequences for
$\rho_{0}$ is the maximal one (i.e., for any other (semi)metric, sequences
$\\{c_{n}\\}$ from the corresponding class grow not faster than for $\rho$).
In symbols, we write this fact as

${\cal H}_{\rho_{0}}(T)=\sup_{\rho}{\cal H}_{\rho}(T).$

Then we say that ${\cal H}_{\rho_{0}}(T)$ is the class of scaling sequences
for the automorphism $T$ and the metric $\rho_{0}$ is $T$-maximal.

It seems that such a metric exists for every automorphism. If we have chosen
some $T$-maximal scaling sequence and the corresponding limit of entropies
does exist, then it is called the scaling entropy.

######  Conjecture 1.

For every ergodic automorphism $T$, a generic $T$-admissible Lipschitz metric
is $T$-maximal. In particular, for a $K$-automorphism (i.e., an automorphism
with completely positive entropy), the scaling sequence is equivalent to the
sequence $c_{n}=h(T)n$, where $h(T)$ is the entropy of $T$, for every
Lipschitz metric.

A preparatory result in this direction was obtained in [17].

In this paper, we will prove that for an automorphism with purely discrete
spectrum and a $T$-admissible metric, the class ${\cal H}_{\rho}(T)$ of
scaling sequences is the class of bounded sequences.

##  5 Invariant metrics on groups and averages of admissible metrics

###  5.1 Invariant metrics and discrete spectrum

Let us recall some known facts about ergodic automorphisms with discrete
spectrum. It obviously follows from the character theory of commutative groups
that the spectrum of a translation on a compact Abelian group is discrete. By
the classical von Neumann theorem, the converse is also true: an ergodic
automorphism with discrete spectrum is metrically isomorphic to the
translation $T$ on a compact Abelian group $G$ endowed with the Haar measure
$m$ by an element whose powers form a dense subgroup:

$x\mapsto Tx=x+g,\quad\operatorname{Cl}\\{ng,\,n\in\mathbb{Z}\\}=G$

(we use the additive notation). Note that on a compact group $G$ there are
many metrics that are invariant under the whole group of translations and
determine the standard group topology. We will need the following assertion
(which is, possibly, partially known).

######  Proposition 2.

The spectrum of an ergodic automorphism $T$ of a measure space $(X,\mu)$ for
which there exists a $T$-invariant admissible semimetric $\rho$ contains a
discrete component. Moreover, if $\rho$ is a metric, then the spectrum of $T$
is discrete and, consequently, $T$ is isomorphic to a translation on a compact
Abelian group.

###### Proof.

Since an admissible (semi)metric lies in the space $L^{1}(X\times
X,\mu\times\mu)$, the tensor square of the operator $U_{T}$, which corresponds
to the automorphism $T\times T$, has nonconstant eigenfunctions. This can
happen only if the spectrum of the unitary operator $U_{T}$ contains a
discrete component, and the first claim is proved. In other words, $T$ is an
extension of some quotient automorphism with discrete spectrum, which may
coincide with $T$ itself. This means that $T$ is a skew product over a base
with discrete spectrum. Denote by $H\subset L^{2}(X,\mu)$ the subspace spanned
by all eigenfunctions of $U_{T}$. All invariant functions of the operator
$U_{T}\otimes U_{T}$ belong to the tensor square $H\otimes H$; hence these
functions, regarded as functions of two variables, do not change when the
argument ranges over a fiber of the skew product. But if these fibers are not
single point sets, it follows that the metric does not distinguish points in
fibers and hence is a semimetric. Thus if $\rho$ is a metric, then each fiber
necessarily consists of a single point, the spectrum of $T$ is purely
discrete, and, by the ergodicity, $T$ is isomorphic to a translation on a
group. ∎

Let us supplement this proof with an important refinement. Assume that the
automorphism $T$ has an orbit that is everywhere dense with respect to the
metric $\rho$, i.e., $T$ is topologically transitive (though we do not assume
that it is a priori continuous). It is clear from above that this condition
follows in fact from the existence of an invariant metric. Since the metric is
admissible, we may assume without loss of generality that $(X,\rho)$ is a
Polish space. Consider the dense orbit $O=\\{T^{n}x,\,n\in\mathbb{Z}\\}$ of
some point $x$. The restriction of $\rho$ to $O$ is a translation-invariant
metric on the group $\mathbb{Z}$, and all translations are isometries. Hence
the completion of $O$ is an Abelian group to which we can extend the
translations and their limits. Therefore $X$ is a Polish monothetic group.555A
topological group that contains a dense infinite cyclic subgroup is called
monothetic. Note that there are many non-locally compact monothetic groups on
which there is an invariant metric, but there is no invariant measure. A
recent example is the Urysohn universal space regarded as a commutative group,
see [9]. Obviously, the measure $\mu$ is invariant under the action of the
closures of powers of $T$, i.e., it is an invariant probability measure on the
whole group. Hence, by Weil’s theorem, $X$ is a compact commutative group and
$T$ is the translation by an element whose powers are everywhere dense.

###  5.2 Admissible invariant metrics

As we have already observed, for weakly mixing automorphisms, the average of
every metric is constant, since there are no other invariant metrics. However,
for automorphims whose spectra contain a discrete component or are purely
discrete, there are many invariant (semi)metrics. Hence, in order to study
such automorphisms, we should investigate the question when the average metric
for an automorphism $T$ with a discrete spectrum is admissible.

Given a translation-invariant metric $\rho$ on a compact commutative group
with the Haar measure, consider the function
$\phi_{\rho}(r)=\rho(x,x+r)=\rho(0,r)$. In the example from Section 3, it
looked as $\phi(r)\equiv\hat{\rho}(x,x+r)=m[A\Delta(A+r)]$.

One can easily write down necessary and sufficient algebraic conditions on a
measurable function $\phi$ that guarantee that it can be written as
$\phi_{\rho}$ for a measurable invariant metrics:

$\phi(\textbf{0})=0,\;\phi(x)\geq
0,\;\phi(-x)=\phi(x),\;\phi(x)+\phi(y)\geq\phi(x+y).$

We will not need these conditions; the only important fact is that the
admissibility condition from Lemma 1 can easily be reformulated in terms of
this function.

######  Theorem 3.

An invariant measurable (semi)metric on a commutative compact group
(satisfying Condition 1 from Definition 1) is admissible if and only if any of
the following conditions holds.

1. The corresponding function $\phi_{\rho}$ is measurable, and

$\mu\\{z:\phi_{\rho}(z)\neq\phi_{\rho}(g+z)\\}>0$

for almost all $g$.

2.

$\operatorname{ess\,inf}_{g\in V\setminus 0}\phi_{\rho}(g)=0,$

where $V$ is an arbitrary neighborhood of the zero of the group in the
standard topology.

###### Proof.

Condition 1 is exactly equivalent to the condition of Lemma 1. It is useful to
give a direct proof that this condition is necessary. Assume that it is not
satisfied, i.e., for all elements $g$ from some set of positive Haar measure,
$\phi_{\rho}(z)=\phi_{\rho}(g+z)$ for almost all $z$. Therefore, the
measurable function $\phi$ is constant on cosets of the subgroup generated by
$g$. However, every set of positive Haar measure in a nondiscrete Abelian
compact group on which there is an ergodic translation contains an element $g$
that generates a dense cyclic subgroup. Then, since the function $\phi_{\rho}$
is measurable, it follows that it is constant almost everywhere. But if the
function $\phi_{\rho}$ is constant, then the metric $\rho$ is also constant
and hence not admissible.

Let us prove that Condition 2 of the theorem is necessary. Consider the
function $\phi_{\rho}$ for an invariant admissible metric $\rho$. Assume that
$\theta=\operatorname{ess\,inf}$ from Condition 2 is positive; then, using the
invariance of $\rho$, we can construct a continuum of points lying at a fixed
positive $\rho$-distance greater than $\theta$, which contradicts the
admissibility. For the same reason, the set of values of $\phi_{\rho}$ is
dense in some neighborhood of the zero. The fact that Condition 2 is
sufficient follows from Proposition 1. ∎

The last assertion implies the following corollary.

######  Corollary 1.

For every admissible invariant metric there exists a sequence of group
elements that converges to zero both in the standard topology and with respect
to the (semi)metric.

###  5.3 Admissibility of the average metric

Now we can explicitly write down the condition that guarantees the
admissibility of the average, i.e., invariant, metric in terms of the original
metric. Consider an arbitrary measurable (non necessarily admissible) metric
$\rho$ on a compact commutative group $G$ and an ergodic translation $T$ by an
element $g$. Let us write down an expression for the average metric:

$\displaystyle{\hat{\rho}}^{T}(x,y)$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n}\rho(x+kg,y+kg)=\int_{G}\rho(g+x,g+y)dm(g)$
$\displaystyle=$ $\displaystyle\int_{G}\rho(z,z+y-x)dm(z)$

(we have used the fact that the measure $m$ is invariant). Hence the function
$\phi(r)$, regarded as a function of $r$, is measurable and has the form

$\phi(r)=\int_{G}\rho(z,z+r)dm(z)=\hat{\rho}(x,y);\quad y-x=r.$

Obviously, $\hat{\rho}$ is a measurable function on the group $G\times G$. Now
we can check the admissibility condition for the average metric $\hat{\rho}$.

######  Definition 7.

We say that an admissible metric $\rho$ on a compact commutative group $G$
with Haar measure $m$ is semicontinuous at zero in mean if

$\liminf_{r\to 0}\int_{G}\rho(x,x+r)dm(x)=0;$

we say that it is semicontinuous at zero in measure if the following condition
holds (in which “meas” means convergence in measure):

$\liminf_{r\to 0}(meas)\rho(x,x+r)=0.$

Note that the second condition follows from the first one, and that both
conditions are stated in purely group terms, i.e., do not depend on the
particular translation $T$.

Thus we have the following admissibility criterion for the average metric.

######  Proposition 3.

The average metric $\hat{\rho}^{T}$ for an ergodic translation $T$ on a
compact commutative group $G$ is admissible if and only if the original
(semi)metric $\rho$ is admissible and semicontinuous at zero in mean.

###### Proof.

The “if” part is proved above; the “only if” part follows from the equality
$\phi_{\hat{\rho}}(r)=\int_{G}\rho(x,x+r)dm(x)$ and the previous proposition.
∎

Now we are ready to formulate and prove the following important fact.

######  Theorem 4.

For every bounded admissible metric on a compact commutative group $G$, the
average metric is admissible.

###### Proof.

Assume that the metric $\rho$ is not semicontinuous at zero and there exists a
positive number $c>0$ such that

$\liminf_{r\to 0}\int_{G}\rho(x,x+r)dm(x)>c.$

Assume that the metric is normalized so that the diameter of the space $X$ is
equal to 1. Then it follows from our assumption that for sufficiently small
(i.e., belonging to a small neighborhood of the zero in the group $G$) $r$
there exists a (depending on $r$) subset in $X$ of measure $\alpha$, which
does not depend on $r$ and is greater than $\frac{c}{2}$, on which
$\rho(x,x+r)>\frac{c}{2}$. Indeed,
$1\cdot\alpha+(1-\alpha)\frac{c}{2}>\int_{G}\rho(x,x+r)dm(x)>c$. But since the
group is compact, there is a set of positive measure for all points $x$ of
which the inequality $\rho(x,x+r)>\frac{c}{2}$ holds for all sufficiently
small $r$ from some set of positive measure. This in turn contradicts the
admissibility of the metric $\rho$ in the formulation of Proposition 1:
arbitrarily small values $\phi(r)$ for small $r$ cannot interlace with values
greater than $\frac{c}{2}$ by the triangle inequality. Thus we have proved
that $\rho$ is semicontinuous. ∎

Question. Does there exist an unbounded admissible metric $\rho$ on the circle
$S^{1}={\mathbb{R}}/{\mathbb{Z}}$ for which the average metric $\hat{\rho}$ is
not admissible? Does there exist an unbounded $p$-admissible metric, with
$1<p<\infty$, on a compact Abelian group for which the average metric is not
admissible?666While the paper was in press, F. Petrov and P. Zatitskiy proved
that the average of any (in particular, unbounded) admissible metric on the
circle is admissible. Thus the additional assumptions that the average metric
is admissible in Proposition 2 and Theorem 5 are superfluous.

##  6 Criterion for the discreteness and continuity of the spectrum in terms
of the scaling entropy

Now we formulate our main result.

######  Theorem 5.

For an ergodic automorphism with discrete spectrum realized as a translation
on a compact commutative group with an arbitrary bounded admissible metric,
or, more generally, with a metric for which the average metric is admissible,
the scaling sequence is bounded.

###### Proof.

Since the set $B_{\varepsilon}=\phi^{-1}([0,\varepsilon])$ is, by definition,
the ball of radius $\varepsilon$ centered at $\textbf{0}\in G$ in the metric
$\hat{\rho}$, which is admissible by assumption, it follows that
$mB_{\varepsilon}>0$, since the metric is nondegenerate (see the definition of
an admissible metric). But the sum $B_{\varepsilon}+B_{\varepsilon}$, like the
sum $A+A$ for every set $A$ of positive Haar measure in a locally compact
group, contains a neighborhood $V$ of the zero in the standard topology (see,
e.g., [28]). It follows from the triangle inequality that
$B_{\varepsilon}+B_{\varepsilon}\subset B_{2\varepsilon}$, so that $V\subset
B_{2\varepsilon}$. Since $\varepsilon>0$ is arbitrary, we see that the
topology on $G$ determined by the average metric $\hat{\rho}$ coincides with
the standard topology, i.e., in the topology determined by $\hat{\rho}$, the
group $G$ is compact and contains a finite $\varepsilon$-net for every
$\varepsilon$.

The pointwise a.e. convergence

$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\rho(x+kg,y+kg)={\hat{\rho}}_{T}(x,y),$

which follows from the pointwise ergodic theorem, implies that the number of
points in an $\varepsilon$-net for $(G,\rho_{n})$ tends to the number of
points in an $\varepsilon$-net for $(G,\hat{\rho})$. This means that the
sequence $H_{\rho_{n},\varepsilon}(X)$ converges to
$H_{\hat{\rho},\varepsilon}(G)$. From the inequalities of Lemma 3 we see that
the sequence $H_{\rho_{n},\varepsilon}(\mu)$ is bounded as $n\to\infty$, and
thus the scaling sequence for the automorphism $T$, which acts on the metric
triple $(G,\rho,m)$, is bounded. ∎

Combining this theorem with the previous one, we obtain the following result.

######  Theorem 6.

An ergodic automorphism $T$ has a discrete spectrum if and only if the scaling
sequence for $T$ is bounded for some, and hence for every, bounded admissible
metric.

###### Proof.

Above we have proved that if an automorphism has a discrete spectrum and the
average metric is admissible, then the scaling sequence is bounded. But the
average metric is always admissible provided that the original metric is
bounded and admissible.

Assume that the scaling sequence is bounded for an automorphism $T$ and an
admissible metric $\rho$. Recall that the average metric is indeed a metric
(and not a semimetric). Consequently, the space $(X,\hat{\rho})$ is
precompact, and hence the metric $\hat{\rho}$ is admissible. Since it is
$T$-invariant, it follows from Theorem 1 that $T$ has a purely discrete
spectrum. ∎

Combining the last theorems with the previous results yields a criterion for
the discreteness and continuity of the spectrum in terms of the automorphism
$T$ and an arbitrary admissible metric.

######  Theorem 7.

Let $T$ be an ergodic automorphism, and let $\rho$ be a bounded admissible
(semi)metric. If the corresponding scaling sequence is not bounded, then the
spectrum of $T$ contains a continuous component. If the scaling sequence is
not bounded for every admissible (semi)metric, then the spectrum of $T$ is
purely continuous.

In the next section, we will show how one could apply this criterion.

##  7 Comparison with the traditional approach, the Pascal automorphism, and
concluding remarks

###  7.1 Supremum metrics

The entropy theory of dynamical systems, developed mainly by Kolmogorov,
Sinai, and Rokhlin, essentially uses the tools of the theory of measurable
partitions. In Sinai’s definition, the entropy appears as an asymptotic
invariant of the dynamics of finite partitions under the automorphism:

$\lim_{n}\frac{H(\bigvee_{k=0}^{n-1}T^{k}\xi)}{n}=h(T,\xi).$

As a result of this theory, the study of the class of automorphisms with
completely positive entropy was differentiated into a separate field, whose
methods do not apply to automorphisms with zero entropy. For example, one
cannot obtain a new invariant for such automorphisms following the same
scheme. This can be seen from the following simple fact.

######  Proposition 4.

For every transformation $T$ and every increasing sequence of positive numbers
$\\{c_{n},\,n\in\mathbb{N}\\}$ satisfying the condition
$lim_{n}\frac{c_{n}}{n}=0$, there exists a generating partition $\xi$ such
that

$\lim_{n}\frac{H(\bigvee_{k=1}^{n}T^{k}\xi)}{c_{n}}=\infty.$

This means that the maximum growth of the entropy
$H(\bigvee_{k=1}^{n}T^{k}\xi)$ either is linear (for automorphisms with
positive entropy), or, in the case it is sublinear, it is arbitrarily close to
linear for every automorphism. Thus we obtain no new information.

The metric corresponding to the supremum (product)
$\xi_{n}=\bigvee_{k=1}^{n}T^{k}\xi$ of partitions is the supremum of the
shifted metrics: ${\bar{\rho}}_{n}^{T}(x,y)=\sup_{0\leq
k<n}\rho(T^{k}x,T^{k}y)$. Hence, following our plan, we can use the
$\varepsilon$-entropy of the metric ${\bar{\rho}}_{n}^{T}(x,y)$ instead of the
entropy of the partition $\xi_{n}$ itself. Then, using the definitions from
Section 4, for a given metric $\rho$ we can introduce an analog of the
function ${\cal H}_{\rho,\varepsilon}(T)$ with the metric $\hat{\rho}_{n}$
replaced by $\bar{\rho}_{n}$:

${\cal\bar{H}}_{\rho,\varepsilon}(T)=\Bigl{\\{}\\{c_{n}\\}:0<\liminf_{n\to\infty}\frac{H_{\bar{\rho}_{n},\varepsilon}(\mu)}{c_{n}}\leq\limsup_{n\to\infty}\frac{H_{\bar{\rho}_{n},\varepsilon}(\mu)}{c_{n}}<\infty\Bigr{\\}}.$

In this way we define the class of $\sup$-scaling sequences $\bar{c}_{n}$ for
a given metric $\rho$. This also allows us to extend the classical entropy
theory following the above scheme. Though it is somewhat easier to deal with
the sup-metric than with the average metric, the former is much less useful
than the latter. The metric $\bar{\rho}$ more often happens to be constant for
an automorphism with discrete spectrum, while, as we have seen, $\hat{\rho}$
is always admissible if the original metric is bounded. Let us illustrate the
important difference between the operations of taking the average and supremum
metrics by the following example.

Example. Let $T$ be an irrational rotation of the unit circle, and let $\rho$
be the semi-metric corresponding to a generating two-block partition (i.e., a
partition into two sets of positive measure); the semi-metric $\rho$ is
$T$-admissible, hence, as we have seen, $\hat{\rho}$ is an invariant
admissible metric. At the same time, $\bar{\rho}$ is the constant metric. This
means that the scaling sequence $c_{n}$ is bounded, but $\bar{c}_{n}$ (the
scaling sequence for the sup-metric) is not; namely, we have
$\bar{c}_{n}\sim\ln n$. Thus the difference manifests itself even in the case
of a discrete spectrum.

Does in make sense to use intermediate averages, e.g., the $l^{p}$-averages

$\lim\Bigl{[}\frac{1}{n}\sum_{k=0}^{n-1}\rho(T^{k}x,T^{k}y)^{p}\Bigr{]}^{\frac{1}{p}}=\hat{\rho}^{p}(x,y)$

for $p\in(1,\infty)$, instead of $l^{1}$? Apparently, they do not lead to any
new effects: these metrics behave in the same way as the $l^{1}$-average
metric. For instance, in the above example, the $p$-average of the metric
determined by a two-block partition of the unit circle is
$\hat{\rho}^{p}(x,y)=\\{m[A\Delta(A+x-y)]\\}^{\frac{1}{p}}$. For $p=\infty$
(i.e., the sup-metric), the picture is completely different, as in other
interpolation theories. Thus in entropy theory, the use of average metrics
substantially supplements the classical considerations.

###  7.2 Application of the discreteness criterion

As noted above, the problem of determining whether or not the spectrum of an
automorphism is discrete, is not at all simple. Theorems 5–7 provide
convenient non-spectral criteria for checking that the spectrum is not purely
discrete; for this, one should bound the entropy from below for one admissible
metric satisfying the conditions of Section 2 by a sequence that grows
arbitrarily slow with $n$.

One of the intriguing examples of automorphisms for which the discreteness of
the spectrum has not been neither proved nor disproved since the 1980s is the
Pascal automorphism. It was introduced by the author in 1980 (see [2, 3]) as
an example of an adic transformation, and is defined as a natural
transformation in the space of paths in the Pascal graph regarded as a
Bratteli–Vershik diagram with lexicographic ordering of paths. One can give a
short combinatorial description of this transformation by encoding these paths
with sequences of zeros and ones and identifying the space of paths with the
compact space $X=\\{0;1\\}^{\infty}=\textbf{Z}_{2}$. Then the Pascal
automorphism is defined by the formula

$T(\\{1^{i}0^{j}1**\\}=\\{0^{j-1}1^{i+1}0**\\});$

here $i\geq 0$, $j>0$, and the domain of $T$ and $T^{-1}$ is the whole $X$
except for the countable set of sequences having finitely many zeros or ones.
The most natural metric on $X$ is the 2-adic metric
$\rho(\\{x_{k}\\},\\{y_{k}\\})=2^{-n}$, where $n$ is the first digit with
$x_{k}\neq y_{k}$. This metric is admissible, and the Pascal automorphism
satisfies the Lipschitz condition almost everywhere. The orbits of this
automorphism coincide with the orbits of the action of the infinite symmetric
group. The Bernoulli measures are $T$-invariant. The spectrum of the Pascal
transformation was studied in the papers [21, 22, 23, 26], where some
interesting properties were established (e.g., it was proved that $T$ is
loosely Bernoulli, the complexity of $T$ was computed, etc.), but the question
about the type of the spectrum remains open. 777Note on the translation: The
answer is known now — the spectrum of Pascal automorphism is continuous. In
[4, 8] it was conjectured that the study of the behavior of scaling sequences
may turn to be useful. The corresponding plan was carried out in [19], but in
that paper a logarithmic lower bound was obtained on the scaling sequence for
the sup-metric, and not for the average metric; this is not sufficient for the
conclusion that the spectrum is not discrete. Nevertheless, one may hope that
the combinatorics developed in [19] will help to prove that the scaling
sequence is unbounded also for the $\varepsilon$-entropy of the average
metric, which, by our theorem, would imply that the spectrum is not discrete.
There are many adic transformations similar to the Pascal automorphism for
which the same question is also of great interest. For example, if we replace
the Pascal graph with its multidimensional analog or the Young graph, we will
obtain automorphisms that supposedly have continuous spectra. As observed
above, in order to prove that there are no nontrivial eigenfunctions, one
should obtain a growing lower bound on the scaling sequence not for one, but
for all (or for some representative set of) bounded admissible (semi)metrics.

###  7.3 The dynamics of metrics

Recall that the general approach that consists in studying the asymptotic
behavior of metrics is not exhausted by considering the asymptotics of the
$\varepsilon$-entropy of the average or supremum metric, i.e., does not reduce
to studying the growth of scaling sequences; this is only its simplest
version. In fact, we consider the original measure space
$(X,{\mathfrak{A}},\mu)$ with an action of an automorphism $T$ (or a group of
automorphisms $G$), fix an appropriate metric $\rho$, and study the sequence
of metric triples

$(X,\rho^{T}_{n},\mu),\quad\mbox{where
}\rho_{n}^{T}(x,y)=\frac{1}{n}\sum_{k=0}^{n-1}\rho(T^{k}x,T^{k}y).$

The conjecture is that, for a fixed measure and a fixed automorphism (or group
of automorphisms), the asymptotic properties of this sequence of metric
triples do not depend (or weakly depend) on the choice of an individual
admissible metric from a wide class. These properties include not only the
scaling entropy, but also more complicated characteristics of the sequence,
say the mutual properties of several consecutive metric triples. Since the
classification of metric triples up to measure-preserving isometry is known
(see [5, 7]), one may hope to apply it to this problem. In this field there
are many traditional and nontraditional questions. For example, what is the
distribution of the fluctuations of the sequence of average metrics, regarded
as functions of two variables on $(X\times X,\mu\times\mu)$, as they converge
to the constant metric (for weakly mixing transformations, e.g.,
$K$-automorphisms)? What can be said about the asymptotic properties of
neighboring pairs of metric triples (with indices $n$ and $n+1$)? Etc.

In conclusion, it is worth mentioning that the concept of scaling entropy
appeared in connection with the classification of filtrations in [14] and was
used in [27]. In terms of the present paper, the scaling entropy for
filtrations, i.e., decreasing sequences of measurable partitions or
$\sigma$-algebras, is the scaling entropy for an action of a locally finite
group such as $\sum{\mathbb{Z}}/2$ instead of an action of $\mathbb{Z}$
considered here. The definitions we have given for an action of $\mathbb{Z}$
essentially coincide with those given in [14] for locally finite groups.

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