# The problem of describing central measures on the path spaces of graded
graphs

A. M. Vershik St. Petersburg Department of Steklov Institute of Mathematics
and St. Petersburg State University. E-mail<EMAIL_ADDRESS>Supported by
the Russian Science Foundation grant 14-11-00581.

(07.08.14)

###### Abstract

We suggest a new method of describing invariant measures on Markov compacta
and path spaces of graphs, and thus of describing characters of some groups
and traces of AF-algebras. The method relies on properties of filtrations
associated with the graph and, in particular, on the notion of a standard
filtration. The main tool is the so-called internal metric that we introduce
on simplices of measures; it is an iterated Kantorovich metric, and the key
result is that the relative compactness in this metric guarantees a
constructive enumeration of the ergodic invariant measures. Applications
include a number of classical theorems on invariant measures.

###### Contents

  1. 1 Introduction
  2. 2 Basic notions
    1. 2.1 A Markov compactum and the path space of a graded graph
    2. 2.2 Additional structures on path spaces
    3. 2.3 Setting of the problem on invariant measures
    4. 2.4 Geometric formulations
    5. 2.5 Extremality of points of a projective limit, and ergodicity of Markov measures
  3. 3 The internal metric and the root norm on projective limits of simplices
    1. 3.1 Metrics on finite sets and the root norm
    2. 3.2 Definition of the internal metric
    3. 3.3 Independence on the original metric
  4. 4 Standardness and uniform compactness
    1. 4.1 Standard extreme points and standard ergodic measures
    2. 4.2 Uniform compactness
  5. 5 Comments and examples
    1. 5.1 Dependence on the approximation and lacunary isomorphism
    2. 5.2 Examples of calculations

##  1 Introduction

The goal of this paper is an exact formulation of a new method of finding
ergodic measures on a Markov compactum or on the path space of a Bratteli
diagram invariant with respect to the tail equivalence relation. The method
presupposes a certain uniform compactness with respect to the metric we
introduce, and if this condition is satisfied, then the complete list of
ergodic invariant measures is parametrized by the points of the
compactification. The ergodicity of a measure in this case means that it
satisfies the 0–1 law with respect to the tail filtration. The suggested
method is an elaboration and strengthening of the ergodic method for finding
the ergodic invariant measures, which is based on the ergodic theorem or the
theorem on convergence of martingales and does not require other conditions.
As a result, it can be applied to any approximation of invariant measures. But
it is this fact that causes serious difficulties when calculating finite-
dimensional distributions. The main difficulty in applying the ergodic method
is not so much in calculations as in the proof that the obtained list is
complete. But it turns out that in many natural situations, such as various
generalizations of de Finetti’s theorem, the theorem on the characters of the
infinite symmetric group, etc., the natural approximation of the tail
partition has much stronger properties than just convergence everywhere or in
measure. Namely, the tail filtration turns out to be standard in the sense of
the theory of filtrations, which means, in particular, that the approximation
converges in a very strong sense. In other words, this can be expressed as
follows: in many examples invariant measures possess latent generalizations of
the independence property, and the standardness exactly takes this adequately
into account, being such a property itself. In fact, the present paper is a
result of applying the theory of filtrations, on which the author has been
working since the 1960s, to problems close to the representation theory of
locally finite groups, their characters and combinatorics. Nevertheless, the
presentation below is self-contained and does not require using other papers.
Here we do not consider the most general graphs, restricting ourselves to what
is needed for the most important examples. A more general class of examples,
requiring additional definitions, will be considered in a paper which is now
in preparation. We only note that the theory of filtrations allows one to look
at finite approximations of invariant measures “from infinity,” i. e., from
the viewpoint of the decreasing sequence of the tail $\sigma$-fields on the
path space, rather than that of the increasing sequence of the finite
$\sigma$-fields of their beginnings.

In Section 2 we give all necessary definitions and a survey of relevant known
facts. The set of invariant (central) measures is a projective limit of
simplices, and its vertices (the Choquet boundary) are the ergodic measures.
That is why we pay attention to the geometry of projective limits, which is
also useful in other problems. The main idea of the paper is described in
Section 3; first we define the Kantorovich metric on a simplex in an
elementary geometric form, and then define the so-called internal metric on a
projective limit of simplices. This notion allows us to divide all ergodic
measures into two classes: standard and nonstandard measures.111In the
preliminary papers [14, 16] we used the term “smooth” and “nonsmooth”
measures, graphs, etc., following the tradition of the theory of
$C^{*}$-algebras, but the terminology suggested here seems to be more
appropriate. The main theorem is given in Section 4: if the levels (pre-limit
simplices) satisfy the uniform compactness condition, then the whole Choquet
boundary consists of the standard ergodic measures and can be found as the
completion of the pre-limit boundaries with respect to the internal metric; in
other words, we obtain a parametrization of the set of ergodic measures and an
interpretation of the parameters. Of course, the generality of this approach
gives no hope for exact formulas for the distributions; they can be (and were)
obtained using the specific features (analytic or probabilistic) in concrete
cases. And though one can hardly hope for this in the general case, a
revelation of the meaning of the parameters and a natural metric on them, and,
above all, the interpretation of their independence, opens a new level in the
understanding of problems concerning invariant measures. In this paper, for
space considerations, we mention examples only briefly in Section 5, meaning
to treat them in another paper.

Figure 1: The Young graph. Figure 2: The graph of unordered pairs.

In conclusion, we comment on the two figures. The first one shows the Young
graph, i. e., the branching graph of irreducible representations of the
infinite group of finite permutations. This graph is standard in the sense of
this paper: all ergodic measures on its path space are standard, and they
comprise the list of indecomposable characters of this group. But proving this
and obtaining the complete list of measures is far from easy. First it was
found analytically by E. Thoma [10]. Later, in [17], the ergodic method idea
was implemented, and the same list was obtained together with an explanation
of the combinatorial and representation-theoretic meaning of the parameters,
which was absent in [10]; even more later, in [7] (see also [6]), this result
was obtained by methods which could be called operator-theoretic. The method
of the present paper is intended to give the first purely combinatorial proof
of the theorem on invariant measures (characters) that would use neither
analytic nor algebraic techniques, but rely only on the combinatorics of the
Young graph. The main point is that the completeness of the list, i. e., the
proof that the frequencies uniquely determine an ergodic measure, a priori
follows from the compactness theorem. And the second graph is the graph of
unordered pairs (the vertices of each level are all possible unordered pairs
of vertices of the previous level), which is an example of a nonstandard
graph. It illustrates in a simplest form the idea of a so-called “tower of
measures” introduced in [13]. Various problems concerning invariant measures
for tame partitions (i. e., monotone limits of finite partitions) can be
reduced to problems on graphs. Examples of nonstandard graphs often arise in
dynamical systems, in the theory of growth of random configurations, in
statistical physics models, etc.

##  2 Basic notions

###  2.1 A Markov compactum and the path space of a graded graph

We will consider the problem of describing the probability measures invariant
with respect to an equivalence relation on the space of trajectories of a
topological Markov compactum or on the space of paths of a graded graph. The
equivalence relation will be given via an approximation by finite equivalence
relations.

The basic space where the measures will be defined is a one-sided topological
Markov compactum ${\cal X}$. Such a compactum is determined by the collection
of state spaces, i. e., by a family of finite sets $\\{X_{n}\\}$,
$n=0,1,\dots$; $X_{0}=\\{\emptyset\\}$, and a family of $0-1$ matrices
$M_{n}=\\{\epsilon_{i,j}\\}$, $i\in X_{n-1}$, $j\in X_{n}$, $n=1,2,\dots$, of
orders $|X_{n-1}|\times|X_{n}|$, $n=1,2,\dots$, without zero rows and columns.
Elements of the Markov compactum ${\cal X}$ are arbitrary infinite sequences
(trajectories) $\\{x_{n}\\}_{n=0,1,\dots}$ of elements of the finite sets
$X_{n}$, $x_{n}\in X_{n}$, $n=1,2,\dots$, satisfying the admissibility
condition which says exactly that every pair of neighboring elements
$x_{n-1},x_{n}$, $n=1,2,\dots$, satisfies $\epsilon_{x_{n-1},x_{n}}=1$. The
topology and the $\sigma$-field of Borel sets in the space ${\cal
X}=\\{\\{x_{n}\\}_{n=0,1,\dots}\\}$ are defined in a natural way, since $\cal
X$ is, obviously, the inverse (projective) limit of the sequence of the finite
spaces ${\cal X}_{n}$ that consist of the trajectories of length $n$
satisfying the same condition with respect to the projections of “forgetting
the last coordinate”: ${\cal X}_{n}\rightarrow{\cal X}_{n-1}$.

Clearly, the above data $\\{X_{n};M_{n}\\}_{n\geq 0}$ determines also another
object: an infinite, locally finite, $\mathbb{N}$-graded graph $\Gamma$, i.
e., a so-called Bratteli diagram. The Markov compactum defined above can be
identified with the space of infinite paths in this graph, which will be
denoted by $T(\Gamma)\sim\cal X$. Bratteli diagrams and their combinatorics
are, in turn, very closely related to the theory of locally semisimple
algebras and AF-algebras, but we will not enter into this here.

Thus we can interpret all that follows not only in terms of Markov compacta,
which are natural for probabilistic statements, but also in terms of graded
graphs $\Gamma$, their levels $\Gamma_{n}\sim X_{n}$, $n=0,1,\dots$, their
vertices $x_{n}\sim\gamma_{n}$, finite or infinite paths $\\{t_{n}=x_{n}\\}$,
etc. Since these two languages coincide almost tautologically (it suffices to
rotate the picture of the graph by $90^{\circ}$ to obtain the scheme of the
Markov chain), in what follows we use both languages simultaneously without
fear of inconsistencies.222There is a trend in terminology, originated in the
theory of the Young graph, in which vertices of the graph $\Gamma$, i. e.,
states of the Markov chain, are called diagrams, paths are called tableaux,
etc.; here we avoid this. The language of graphs is used mainly for the
combinatorial analysis of applications, which have been in fact the main
source of the problems under discussion for the author. Moreover, it turned
out that the arising examples, coming from the theory of Bratteli diagrams,
are poorly studied in the framework of Markov compacta, dynamical systems, and
random processes; and, vice versa, useful constructions from the theory of
dynamical systems only now start finding adequate applications in the
algebraic theory.

######  Remark 1.

It is convenient to adopt the following convention: no two distinct vertices
of the same level of the graph (respectively, no two elements of the state
space of the Markov compactum) have the same set of preceding vertices (i. e.,
the matrices $M_{n}$ have no identical rows). In what follows, this condition
will ensure that the internal metric is nondegenerate. It is not too
restrictive, since if there are such vertices, then we can replace them by a
single vertex and change the multiplicities of edges by multiplying them with
the number of merged vertices; applying this procedure to all vertices, we
obtain a new graph satisfying the required condition.

The most important notions for what follows are the tail equivalence relation
on a Markov compactum, or on the path space of a graph, and the tail
filtrations on these spaces. Two paths, or two sequences of elements in a
Markov compactum, are called $n$-cofinal (equivalent) if they coincide from
the $n$th level; the $n$-cofinality equivalence classes are finite. The
measurable (defined as the partition into the preimages of a Borel-measurable
function) partition of the path space into these classes will be denoted by
$\xi_{n}$. Two paths are called cofinal if they are $n$-cofinal for some $n$.
The cofinality classes determine an equivalence relation which will be called
the tail equivalence relation and denoted by $\xi(\cal X)$ or $\xi(\Gamma)$.
In general, the partition into the cofinality classes, i. e., the intersection
of the $n$-cofinality partitions $\xi_{n}$ over all $n$, is not measurable,
but it is hyperfinite, i. e., by definition, is a limit of Borel equivalence
relations with finite equivalence classes. The $\sigma$-field of
$\xi_{n}$-measurable sets (i. e., sets that along with a given path contain
all paths $\xi_{n}$-equivalent to it) will be denoted by ${\mathfrak{A}}_{n}$.
The $\sigma$-fields ${\mathfrak{A}}_{n}$ decrease with $n$, the first of them
coinciding with the whole original $\sigma$-field:
${\mathfrak{A}}_{0}=\mathfrak{A}$. Decreasing sequences of $\sigma$-fields in
measure spaces or topological spaces are called filtrations; the above
sequence $\\{{\mathfrak{A}}_{n}\\}$ will be called the tail filtration on the
path space of the graph or on the Markov compactum. These notions and their
properties, which have been studied since the 1970s (see [11, 13]), play an
important role in this paper.

###  2.2 Additional structures on path spaces

On the objects described above one can define important additional structures.
We will consider them later, and now give only a short description.

1. The finite set of edges leading to a given vertex can be endowed with a linear ordering; if it is chosen for all vertices, this allows one to define a lexicographic ordering on the classes of cofinal paths, and then introduce a partial transformation of paths that sends a given path to the next one in the lexicographic ordering, the so-called adic shift. Here we will not use this notion; the only useful observation is that the tail equivalence relation is nothing but the trajectory partition for the adic shift.

2. Another additional structure is the cocycle of cotransition probabilities: with each vertex $\gamma_{n}\in\Gamma$ (or coordinate $x_{n}\in X_{n}$) one associates the probability vector $\\{\lambda_{\gamma_{n}}^{\gamma_{n-1}}\\}$ where $\gamma_{n-1}$ ranges over all vertices from $\Gamma_{n-1}$ that are connected with $\gamma_{n}$. This vector is interpreted as the vector of cotransition probabilities of the Markov chain, i. e., as $\mbox{Prob}\\{x_{n-1}|x_{n}\\}$.

In this paper we will consider this structure only in the basic special case
canonically associated with the graph and the Markov compactum. Namely, the
cotransition probability $\lambda_{\gamma_{n}}^{\gamma_{n-1}}$ (for a pair
$(\gamma_{n-1},\gamma_{n})$) is proportional to the number of paths leading to
$\gamma_{n-1}$ from the initial vertex. Measures with such cotransition
probabilities are called central (invariant), see below. However, the method
suggested in this paper applies without changes to the case of general
cotransition probabilities (i. e., to the theory of quasi-invariant measures).
The number of paths leading from $\emptyset$ to a vertex $\gamma$ is usually
denoted by $\dim\gamma$ (since this is the dimension of a certain module).

3. The matrices $M_{n}$ determining a Markov compactum may have nonnegative integer entries, which corresponds to the case of a multigraph (a graph with multiple edges); in this case, a path is a sequence of edges rather than vertices. All main theorems remain valid in this case. One may consider an even more general context, where multiplicities are nonnegative reals corresponding to weights of edges and paths.

Note that all additional structures introduced above can be rephrased in terms
of generalized matrices $M_{n}$ and interpreted in terms of locally semisimple
algebras and AF-algebras associated with Bratteli diagrams; we will not enter
into these questions here.

###  2.3 Setting of the problem on invariant measures

We turn to describing the main problem.

Assume that we are given a Markov compactum $\cal X$ (or the path space
$T(\Gamma)$ of a Bratteli diagram); the set $\mbox{Meas}(\cal X)$ of all Borel
probability measures on $\cal X$ is an affine compact (in the weak topology)
simplex, whose extreme points are delta measures. Since $\cal X$ is an inverse
(projective) limit of finite spaces (namely, the spaces of finite paths), it
obviously follows that $\mbox{Meas}(\cal X)$ is also an inverse limit of
finite-dimensional simplices $\hat{\Sigma}_{n}$, where $\hat{\Sigma}_{n}$ is
the set of formal convex combinations of finite paths (or just the set of
probability measures on these paths) leading from the initial vertex to
vertices of level $n$, $n=1,2,\dots$, and the projections
$\hat{\pi}_{n}:\hat{\Sigma}_{n}\to\hat{\Sigma}_{n-1}$ correspond to
“forgetting” the last vertex of a path. Every measure is determined by its
finite-dimensional projections to cylinder sets (i. e., is a so-called
cylinder measure). We will be interested only in invariant (central) measures,
which form a subset of $\mbox{Meas}(\cal X)$.

######  Definition 1.

A Borel probability measure $\mu$ on a Markov compactum (= on the path space
of a graph) is called central if for any vertex of an arbitrary level, the
projection of this measure to the subfield of cylinder sets of finite paths
ending at this vertex is the uniform measure on this (finite) set of paths.

Other, equivalent, definitions of a central measure $\mu\in\mbox{Meas}(\cal
X)$ are as follows.

1. The conditional measure of $\mu$ obtained by fixing the “tail” of infinite paths passing through a given vertex, i. e., the conditional measure of $\mu$ on the elements of the partition $\xi_{n}$, is the uniform measure on the initial segments of paths for any vertex.

2. The measure is invariant under any adic shift (for any choice of orderings on the edges).

3. The measure is invariant with respect to the tail equivalence relation.

The term “central measure” stems from the fact that in the application to
representation theory of algebras and groups, measures with these properties
determine traces on algebras (respectively, characters on groups). In the
theory of stationary (homogeneous) topological Markov chains, central measures
are called measures of maximal entropy.

The set of central measures on a Markov compactum $\cal X$ (on the path space
$T(\Gamma)$ of a graph $\Gamma$) will be denoted by $\operatorname{Inv}({\cal
X})$ or $\operatorname{Inv}(\Gamma)$. Clearly, the central measures form a
convex weakly closed subset of the simplex of all measures:
$\operatorname{Inv}({\cal X})\subset\operatorname{Meas}(\cal X)$. The set
$\operatorname{Inv}({\cal X})$ of central measures is also a simplex, which
can be naturally presented as a projective limit of the sequence of finite-
dimensional simplices of convex combinations of uniform measures on the
$n$-cofinality classes. In more detail:

######  Proposition 1.

The simplex of central measures can be written in the form

$\operatorname{Inv}({\cal X})=\lim_{\leftarrow}(\Sigma_{n};p_{n,m}),$

or

$\Sigma_{1}\leftarrow\Sigma_{2}\leftarrow\dots\leftarrow\Sigma_{n}\leftarrow\Sigma_{n+1}\leftarrow\dots\leftarrow\Sigma_{\infty}\equiv\operatorname{Inv}({\cal
X}),$

where $\Sigma_{n}$ is the simplex of formal convex combinations of vertices of
the $n$th level $\Gamma_{n}$ (i. e., points of $X_{n}$), and the projection
$p_{n,n-1}:\Sigma_{n}\to\Sigma_{n-1}$ sends a vertex $\gamma_{n}\in\Gamma_{n}$
to the convex combination
$\sum\lambda_{\gamma_{n-1}}^{\gamma_{n}}\delta_{\gamma_{n-1}}\in\Sigma_{n-1}$
where the numbers $\lambda_{\gamma_{n-1}}^{\gamma_{n}}$ are uniquely
determined by the condition that $\lambda_{\gamma_{n-1}}^{\gamma_{n}}$ is
proportional to the number of paths leading from $\emptyset$ to $\gamma_{n-1}$
(which is denoted, as already mentioned, by $\dim\gamma_{n-1}$).333In the
general (noncentral) case, the coefficients $\lambda$ are the cotransition
probabilities (see above). The general form of the projection is
$p_{n,m}=\prod_{i=m}^{n+1}p_{i,i-1}$, $m>n$.

###### Proof.

The set of all Borel probability measures on the path space is a simplex which
is a projective limit of the simplices generated by the spaces of finite paths
of length $n$ in the graph, which follows from the fact that the path space
itself is a projective limit with the obvious projections of “forgetting” the
last edge of a path. The space of invariant measures is thus a weakly closed
subset of this simplex, and we will show that it is also a projective limit of
simplices (the fact that it is a simplex is well known, see, e.g., [8]). The
projection $\mu_{n}$ of any invariant measure $\mu$ to a finite cylinder of
level $n$ is a measure invariant under changes of initial segments of paths
and hence lies in the simplex defined above; since the projections preserve
this invariance, $\\{\mu_{n}\\}$ is a point of the projective limit. It
remains to observe that a measure is uniquely determined by its projections,
which establishes a bijection between the points of the projective limit and
the set $\operatorname{Inv}(\Gamma)$ of invariant measures. ∎

Recall that points of the simplex $\Sigma_{n}$ are probability measures on the
points of $X_{n}$ (i.e., on the vertices of the $n$th level $\Gamma_{n}$), and
the extreme points of $\Sigma_{n}$ are exactly these vertices. Remark 1 means
that distinct vertices of the graph correspond to distinct vertices of the
simplex.

Extreme points of the simplex $\operatorname{Inv}(\Gamma)$ of invariant
measures on the whole path space $T(\Gamma)$ are indecomposable invariant
measures, i. e., measures that cannot be written as nontrivial convex
combinations of other invariant measures. Then it follows from the theorem on
the decomposition of measures invariant with respect to a hyperfinite
equivalence relation into ergodic components that an indecomposable measure is
ergodic ( = there are no invariant subsets of intermediate measure). It is
these measures that are of most interest to us, since the other measures are
their convex combinations, possibly continual. The set of ergodic central
measures of a Markov compactum $\cal X$ (of a graph $\Gamma$) will be denoted
by $\mbox{Erg}(\cal X)$ or $\mbox{Erg}(\Gamma)$.

######  Problem 1.

Describe all central ergodic measures for a given Markov compactum
(respectively, all indecomposable central measures for a given graph). A
meaningful question is for what Markov compacta or graphs the set of ergodic
central measures has an analytic description in terms of combinatorial
characteristics of this compactum or graph, and what are these
characteristics; and in which cases such a description does not exist. The
role of such characteristics may be played by some properties of the sequence
of matrices $\\{M_{n}\\}$ determining the compactum (graph), frequencies,
spectra, etc.

This problem includes those of describing unitary factor representations of
finite type of discrete locally finite groups, finite traces of some
$C^{*}$-algebras, Dynkin’s entrance and exit boundaries (see [1]); it is very
closely related to the problems of finding Martin boundaries,
Poisson–Furstenberg boundaries, etc. The answer to the question stated in
Problem 1 may be either “tame” (there exists a Borel parametrization of the
ergodic measures or the factor representations of finite type) or “wild” (such
a parametrization does not exist). As is well known since the 1950s, in the
representation theory such is the state of affairs in the theory of
irreducible representations of groups and algebras. However, this also
happens, though more rarely, in the theory of factor representations. But in
many classical situations the answer in “tame,” which is a priori far from
obvious.

For example, the characters of the infinite symmetric group, i.e., the
invariant measures on the path state of the Young graph (see Fig. 1), have a
nice parametrization, and this is a deep result; however, for the graph of
unordered pairs (see Fig. 2) there is no nice parametrization. We emphasize
that the presentation of $\operatorname{Inv}(\Gamma)$ as a projective limit of
simplices relies essentially on the approximation, i. e., on the structure of
the Markov compactum (graph). Obviously, the answer to the stated question
also depends on the approximation. The fact is that we can change the
approximation without changing the stock of invariant measures, which is
determined only by the tail equivalence relation. The dependence of our
answers on the approximation will be discussed later (see the remark on the
lacunary isomorphism theorem in the last section). But since in actual
problems the approximation is explicit already in the setting of the problem,
the answer should also be stated in its terms. See examples below.

###  2.4 Geometric formulations

We will recall some well known geometric formulations, since the language of
convex geometry is convenient and illustrative in this context.

1\. The set of all Borel probability measures on a separable compact set
invariant under the action of a countable group (or equivalence relation) is a
simplex (= Choquet simplex, see [8]), i. e., a separable affine compact set in
the weak topology whose any point has a unique decomposition into an integral
with respect to a measure on the set of extreme points.444Choquet’s theorem on
the decomposition of points of a convex compact set into an integral with
respect to a probability measure on the set of extreme points is a
strengthening, not very difficult, of the previous fundamental Krein–Milman
theorem saying that a convex affine compact set is the weak closure of the set
of convex combinations of extreme points. The set of ergodic measures is the
Choquet boundary, i. e., the set of extreme points, of this simplex; it is
always a $G_{\delta}$ set.

2\. Terminology (somewhat less than perfect): a Choquet simplex is called a
Poulsen simplex [9] if its Choquet boundary is weakly dense in it, and it is
called a Bauer simplex if the boundary is closed. Cases intermediate between
these two ones are possible.

3\. A projective limit of simplices (see below) is a Poulsen simplex if and
only if for any $n$ the union of the projections of the vertex sets of the
simplices with greater numbers to the $n$th simplex is dense. The universality
of a Poulsen simplex was observed and proved much later [2, 3]:

######  Proposition 2.

All separable Poulsen simplices are topologically isomorphic as affine
compacta; this unique, up to isomorphism, simplex is universal in the sense of
model theory.555That is, every separable simplex can be mapped injectively
into the Poulsen simplex, and an isomorphism of any two isomorphic faces of
the Poulsen simplex can be extended to an automorphism of the whole simplex.

One can easily check that every projective limit of simplices arises when
studying quasi-invariant measures on the path space of a graph, or Markov
measures with given cotransition probabilities (see above). But in what
follows we consider only central measures, i. e., take a quite special system
of projections in the definition of a projective limit. However, there is no
significant difference in the method of investigating the general case
compared with the case of central measures. We will return to this question
elsewhere.

We add another two simple facts, which follow from definitions.

4\. Every ergodic central measure on a Markov compactum (on the path space of
a graph) is a Markov measure with respect to the structure of the Markov
compactum (the ergodicity condition is indispensable here).

5\. The tail filtration is semi-homogeneous with respect to every ergodic
central measure, which means exactly that almost all conditional measures for
every partition $\xi_{n}$, $n=1,2,\dots$, are uniform.

The metric theory of semi-homogeneous filtrations will be treated in a
separate paper.

###  2.5 Extremality of points of a projective limit, and ergodicity of Markov
measures

We give a criterion for the ergodicity of a measure in terms of general
projective limits of simplices, in other words, a criterion for the
extremality of a point of a projective limit of simplices.

Assume that we are given an arbitrary projective limit of simplices
$\Sigma_{1}\leftarrow\Sigma_{2}\leftarrow\dots\leftarrow\Sigma_{n}\leftarrow\Sigma_{n+1}\leftarrow\dots\leftarrow\Sigma_{\infty}$
with affine projections $p_{n,n-1}:\Sigma_{n}\rightarrow\Sigma_{n-1}$,
$n=1,2,\dots$ (the general projection
$p_{m,n}:\Sigma_{m}\rightarrow\Sigma_{n}$ is given above).

Consider an element $x_{\infty}\in\Sigma_{\infty}$ of the projective limit; it
determines, and is determined by, the sequence of its projections
$\\{x_{n}\\}_{n=1,2,\dots}$, $x_{n}\in\Sigma_{n}$, to the finite-dimensional
simplices. Fix positive integers $n<m$ and take the (unique) decomposition of
the element $x_{m}$, regarded as a point of the simplex $\Sigma_{m}$, into a
convex combination of its extreme vertices $e_{i}^{m}$:

$x_{m}=\sum_{i}c_{m}^{i}\cdot e_{i}^{m},\quad\sum_{i}c_{m}^{i}=1,\quad
c_{m}^{i}\geq 0;$

denote by $\mu_{m}=\\{c_{m}^{i}\\}_{i}$ the measure on the vertices of
$\Sigma_{m}$ corresponding to this decomposition. Project this measure
$\mu_{m}$ to the simplex $\Sigma_{n}$, $n<m$, and denote the obtained
projection by $\mu_{m}^{n}$; this is a measure on $\Sigma_{n}$, and thus a
random point of $\Sigma_{n}$; note that this measure is not in general
concentrated on the vertices of the simplex $\Sigma_{n}$.

######  Proposition 3 (Extremality of a point of a projective limit of
simplices).

A point $x_{\infty}=\\{x_{n}\\}_{n}$ of the limit simplex $\Sigma_{\infty}$ is
extreme if and only if the sequence of measures $\mu_{n}^{m}$ weakly
converges, as $m\to\infty$, to the delta measure $\delta_{x_{n}}$ for all
values of $n$:

$\displaystyle\mbox{for every }\epsilon>0,\mbox{ for every }n\mbox{ there
exists }K=K_{\epsilon,n}\mbox{ such that }$
$\displaystyle\mu_{n}^{m}(V_{\epsilon}(\mu_{n}))>1-\epsilon\quad\mbox{for
every }m>K,$

where $V_{\epsilon}(\cdot)$ is the $\epsilon$-neighborhood of a point in the
usual (for instance, Euclidean) topology.

It suffices to use the continuity of the decomposition of an arbitrary point
$x_{\infty}$ into extreme points in the projective limit topology, and project
this decomposition to the finite-dimensional simplices; then for extreme
points, and only for them, the sequence of projections must converge to a
delta measure.

One can easily rephrase this criterion for our case
$\Sigma_{\infty}=\operatorname{Inv}(\Gamma)=\operatorname{Inv}(\cal X)$. Now
it is convenient to regard the coordinates (projections) of a central measure
$\mu_{\infty}$ not as points of finite-dimensional simplices, but as measures
$\\{\mu_{n}\\}_{n}$ on their vertices (which is, of course, the same thing).
Then the measures $\mu_{m}^{n}$ should be regarded as measures on probability
vectors indexed by the vertices of the simplex, and the measure $\mu$ on the
Markov compactum $\cal X$ (or on $T(\Gamma)$), as a point of the limit simplex
$\operatorname{Inv}$. The criterion then says that $\mu$ is an ergodic measure
(i. e., an extreme point of $\operatorname{Inv}$) if and only if the sequence
of measures $\mu_{m}^{n}$ (on the set of probability measures on the vertices
of the simplex $\Sigma_{n}$) weakly converges as $m\to\infty$ to the measure
$\mu_{n}$ (regarded as a measure on the vertices of $\Sigma_{n}$) for all $n$.

In probabilistic terms, our assertion is a topological version of the theorem
on convergence of martingales in measure and has a very simple form: for every
$n$, the conditional distribution of the coordinate $x_{n}$ given that the
coordinate $x_{m}$, $m>n$, is fixed converges in probability to the
unconditional distribution of $x_{n}$ as $m\to\infty$.

According to this proposition, in order to find the finite-dimensional
projections of ergodic measures, one should enumerate all delta measures that
are weak limits of measures $\mu_{n}^{m}$ as $m\to\infty$. But, of course,
this method is inefficient and tautological. The more efficient ergodic method
requires, in order to be justified, a strengthening of this proposition,
namely, replacing convergence in measure with convergence almost everywhere,
i. e., the individual ergodic theorem, or pointwise convergence of martingales
(see [12, 17]). The point of this paper is the enhancement of the ergodic
method via a new type of convergence of martingales.

##  3 The internal metric and the root norm on projective limits of simplices

###  3.1 Metrics on finite sets and the root norm

We begin with the following simple remark, which however plays a fundamental
role. Namely, we give an elementary definition of the Kantorovich metric in
the simplest finite-dimensional case in a purely geometric framework. Consider
a finite set $E=\\{e_{i}\\}$ and construct the finite-dimensional simplex
$\Sigma(E)$ of formal convex combinations of points of this set. Assume that
on the set $E$, i. e., on the set of vertices $\\{e_{i}\\}$ of the simplex
$\Sigma(E)$, there is a metric $\rho$. This metric can be extended to each of
the edges of $\Sigma(E)$ as the Euclidean metric such that the length of the
edge is equal to the distance between its endpoints. Is there a natural way to
extend it to higher-dimensional faces and to the whole simplex, and what is
the stock of such extensions?

######  Theorem 1 (Definition of the extension of a metric).

There exists a functorial extension $\bar{\rho}$ of a metric $\rho$ defined on
the vertex set of a simplex to the whole simplex. We mean that the extension
procedure described below is a functor from the category of finite metric
spaces to the category of affine simplices of a real vector space with a
metric.

This extension is nothing but the Kantorovich (or transportation) metric on
the simplex; it determines a norm in the affine hull of the simplex (the
Kantorovich–Rubinshtein norm), which is maximal in the class of all possible
extensions of the metric from the set of vertices to the affine hull.

###### Proof.

The extension of the metric and the norm are defined as follows.

With every point $x\in\Sigma$ of the simplex we can associate the unique
probability measure $\nu_{x}$ on the set of vertices $e_{i}$, $i=1,2,\dots,n$,
of $\Sigma$ whose barycenter coincides with $x$:

$x=\sum_{i}c_{i}e_{i};\quad\nu_{x}=\\{c_{i}\delta_{e_{i}}\\};\quad c_{i}\geq
0,\quad\sum_{i}c_{i}=1.$

Consider the Kantorovich distance $k_{\rho}$ between two measures $\nu_{x}$
and $\nu_{y}$ corresponding to points $x$ and $y$ of the simplex; it is
declared to be the extension $\bar{\rho}$ of the metric $\rho$ to the whole
simplex. We give the definition of the Kantorovich metric in our notation:

$\bar{\rho}(x,y)\equiv
k_{\rho}(\nu_{x},\nu_{y})=\min_{\psi}\left\\{\sum_{i,j}\psi_{i,j}\rho(e_{i},e_{j}):\psi=\\{\psi_{i,j}\\},\sum_{j}\psi_{ij}=c_{i},\sum_{i}\psi_{ij}=d_{j}\right\\}.$

One can easily see that on the vertex set of $\Sigma$, the metric $\bar{\rho}$
coincides with the original metric:
$\bar{\rho}(e_{i},e_{j})=\rho(e_{i},e_{j})$. On the edges of $\Sigma$, it
coincides with the usual Euclidean metric with given distances between the
endpoints of edges. However, on higher-dimensional faces, it depends
substantially on the original metric on the vertices. It is convenient to
embed the simplex into its affine hull and choose the sum of its vertices as
the origin of the linear space. Then the metric on the simplex determines a
(Kantorovich–Rubinshtein) norm on the affine hull, which is given by the
formula

$\|c-c^{\prime}\|=k_{r}(c,c^{\prime}),\quad\|\lambda c\|=|\lambda|\|c\|,$

where $c,c^{\prime}$ are two points of the cone spanned by the simplex, and
this norm is well defined since it does not depend on the representation of an
element as the difference of points of the cone.

The maximality of this norm in the class of all norms on the affine hull for
which $\|\delta_{x}-\delta_{y}\|=\rho(x,y)$ follows from the fact that the
unit ball in this norm is the convex hull of the elements of the form
$\delta_{x}-\delta_{y}$ (the simple roots, see below).666Maximality is not the
only property of the Kantorovich metric that singles it out from the set of
all possible extensions of a metric from the vertex set of a simplex to the
whole simplex; another such property is used in Proposition 5. ∎

Thus there is a finitely parametrized family of norms in $\mathbb{R}^{n}$ each
of which is determined by its values on the differences of coordinate vectors.
This is exactly the family of Kantorovich metrics and Kantorovich–Rubinshtein
norms in the simplest (finite) case of the transportation problem. The
Kantorovich–Rubinshtein norm and its dual (Lipschitz) norm have been studied
in many papers in the infinite-dimensional, continual, case; however, the
geometry of balls in the above norms (i. e., in the finite-dimensional spaces
$\mathbb{R}^{n}$) is not less interesting than the geometry of balls and norms
in $l^{p}$ or other traditional norms. Strange as it may seem, these norms are
not mentioned in textbooks on optimization theory or convex geometry, as far
as the author knows. In this relation, see the paper [4] devoted to an
entirely different subject.

The unit balls in these metrics are very interesting convex centrally
symmetric polyhedra. In the dimension $d=2$, this ball is an arbitrary
centrally symmetric hexagon, i.e., the norm is hexagonal; for $d=3$, it is a
polyhedron combinatorially equivalent to a cube. The general case is more
complicated and, apparently, has not been studied. The metric $\bar{\rho}$ on
the simplex $\Sigma$ can also be called the root metric corresponding to the
metric $\rho$ on the vertices of $\Sigma$. If $\rho=\delta_{x,y}=1$ for $x\neq
y$, this is the root metric in the sense of the Lie algebra
${\widehat{A}}_{n}$.777This norm is related to the theory of Lie algebras in
the following way. Assume that our simplex lies in the Cartan subalgebra of
the Lie algebra ${\hat{A}}_{n}$ and is generated by its simple roots, on which
the delta metric is considered. Then the unit ball in this norm is the convex
hull of all positive and negative simple roots; it is natural to call it the
root norm. The root norm is invariant under the Weyl group; has it been
considered in detail (see [18])?

######  Remark 2.

The definition immediately implies an important monotonicity property of the
metrics under consideration:

$\bar{\rho}_{n-1}(p_{n,n-1}x,p_{n,n-1}y)\leq\bar{\rho}_{n}(x,y),\quad
x,y\in\Sigma_{n}.$

###  3.2 Definition of the internal metric

We pass to an inductive construction of our key notion, that of the internal
metric on projective limits of simplices and, in particular, on the simplex of
invariant measures of a Markov chain. For this we iterate the Kantorovich
metric with growing dimension. Assume that we are given a projective limit
$\lim_{n}\\{\Sigma_{n},p_{n,m}\\}$ of simplices.

######  Definition 2.

Fix an arbitrary metric $\rho_{k}$ on the vertex set of the simplex
$\Sigma_{k}$ for some $k\geq 1$ and its extension to the whole simplex
$\Sigma_{k}$, which we will now denote by the same symbol $\rho_{k}$. We
define metrics on all the subsequent simplices $\Sigma_{n}$, $n\geq k$, by
induction as follows. By the definition of a projective limit, the vertices of
$\Sigma_{n+1}$, $n\geq k$, are projected to some points of the previous
simplex $\Sigma_{n}$, on which the metric is already defined. Take it as a
metric on the vertices of $\Sigma_{n+1}$ and extend it as described above to a
metric on the whole simplex $\Sigma_{n+1}$, which will also be denoted by
$\rho_{n+1}$.

In what follows, we assume that the original metric $\rho_{k}$ is fixed and
denote the iterated internal metrics by $\\{\rho_{n}\\}_{n=1}^{\infty}$; below
we will show that the key property (standardness) does not depend on the
choice of this metric.888Digressing for a moment from the main line, observe
that the internal metric defined above has the following interpretation: it is
a metric on the space of graded finite-dimensional modules, since this is
exactly what the vertices of a Bratteli diagram are. This interpretation is of
importance for algebraic applications. It would be interesting to compare it
with other metrics on the spaces of modules of algebras if they exist.

We have obtained a family of metrics, nondecreasing with $n$, on each of the
simplices and, in particular, on the collection of all their vertices. We will
show how to extend them to the disjoint union of the vertices of all
simplices. First we define the distance between a vertex $x\in\Sigma_{n}$ and
a vertex $f\in\Sigma_{n+1}$ as the distance between $x$ and the projection of
$f$ to $\Sigma_{n}$ in the internal metric on $\Sigma_{n}$. Further, the
distance between vertices $e\in\Sigma_{n}$ and $f\in\Sigma_{m}$, $m>n$, is
defined as the minimum of the sums of the distances $\rho(e_{i},e_{i+1})$ over
all pairs of vertices of neighboring levels for all chains of vertices
$e=e_{n},e_{n+1},\dots,e_{m-1},e_{m}=f$ of intermediate levels with final
point $f$ and initial point $e$. One could also define a metric on the whole
union of the simplices, but we will not need it.

The definition of the internal metric on the vertices of a simplex allows us
to translate it to the vertices of a graph and to the states of a Markov
chain; by Remark 1, every vertex of the simplex is associated with only one
vertex of the graph, hence the image of the metric on the vertices of the
graph is a metric (rather than a semimetric).

######  Remark 3.

It is natural to give a general definition of sequences of internal metrics
(or semimetrics) not for the vertices of simplices or graphs, but for the
space of sequences of vertices (edges) of a graph, i. e., for the path space
of the graph or the trajectory space of the Markov chain. This can be done by
the universal trick of “translating a semimetric” similar to that described
above. The iterations will result in a semimetric on the space of infinite
paths, which can be quotiened to obtain a metric on the space of central
measures, or, more exactly, on the space of disjoint Borel (but, in general,
nonclosed) supports of ergodic central measures. Since in many applications it
suffices to regard internal metrics as already defined and compactify the
union of the vertex sets, we postpone the general definition till another
paper.

###  3.3 Independence on the original metric

The fact that the standardness of an extreme point does not depend on the
original metric used to construct the internal metric is implied by the
following result.

######  Proposition 4.

Let $S$ be an arbitrary finite set, and let $\rho,\rho^{\prime}$ be two
metrics on $S$ such that

$\rho(x,y)\leq r\cdot\rho^{\prime}(x,y)$

for all $x,y\in S$, where $r$ is a positive constant. Then the corresponding
Kantorovich metrics on the simplex $\Sigma(S)$ of measures satisfy the similar
inequality:

$r_{\rho}(\mu_{1},\mu_{2})\leq r\cdot r_{\rho^{\prime}}(\mu_{1},\mu_{2}).$

###### Proof.

We make use of the duality theorem that presents the value of the Kantorovich
metric as the supremum over the unit ball of the integrals of Lipschitz
functions with respect to the difference of the measures:

$r_{\rho}(\mu_{1},\mu_{2})=\sup_{u\in\mbox{Lip}_{\rho,1}}\int_{S}u(x)d(\mu_{1}-\mu_{2}).$

Obviously, the inequality for the metrics implies the following relation
between the unit balls of the spaces of Lipschitz functions:

$\mbox{Lip}_{\rho,1}\subset r\cdot\mbox{Lip}_{\rho^{\prime},1},$

and the required inequality follows. ∎

######  Corollary 1.

The asymptotic (with respect to the dimension) properties of the internal
metrics, such as convergence of sequences etc., do not depend on the original
metric.

Indeed, by the proposition, inequalities between metrics are preserved under
iterations.

In the general case, the original metric is defined on the space of paths
(finite or not) starting at the zero-level vertex, and successive iterations
translate it to paths starting from the first, second, etc. level (see Remark
3). The conclusion and the proof that the limit behavior of these metrics does
not depend on the original one remain valid in this case, too.

##  4 Standardness and uniform compactness

###  4.1 Standard extreme points and standard ergodic measures

A trivial extreme point of a projective limit of simplices is a sequence
$\\{x_{n}\\}$ consisting of extreme points of the pre-limit simplices:
$x_{n}\in\mbox{Ex}(\Sigma_{n})$; such points are of no interest even if they
exist. On the other hand, some extreme points of projective limits satisfy
much stronger concentration conditions than in the extremality condition
(Proposition 4). The case of importance for us is where the sequence of
projections $\\{x_{n}\\}$ is such that their distances to the set of vertices
of the simplex in some metric tend to zero. In order to state exactly what
such a convergence means, we must introduce a metric on the simplices of all
dimensions according to some common rule.999This convergence can be universal
if the definition is functorial. Just such a definition was given above: this
is the internal metric.

This allows us to divide the set of extreme points, and, in particular, the
set of ergodic central measures on a Markov compactum and on the path space of
a graph, into two classes.

Because of the importance of this definition, we first state it separately for
an arbitrary projective limit of simplices, and then specialize for measures
on Markov compacta and on the path spaces of graphs. It is not difficult to
see that these statements express the same property in different
terms.101010In this paper, we do not enter into the question of how all these
formulations are related to the general definition of a standard filtration,
neither to the question why the standardness in the sense of these definitions
is a property of a filtration that does not depend on the way in which it is
realized as a tail filtration. These questions will be studied in the above-
mentioned paper in preparation on the metric and combinatorial theory of
filtrations.

######  Definition 3.

1. Consider a projective limit of simplices $\lim\\{\Sigma_{n},p_{m,n}\\}\equiv\Sigma_{\infty}$. An extreme point $x_{\infty}\in\Sigma_{\infty}$ with projections $\\{x_{n}\in\Sigma_{n}\\}$ is called standard if $\lim_{n\to\infty}\rho_{n}(x_{n},\mbox{\rm Ex}(\Sigma_{n}))=0$, i. e., the distance from $x_{n}$ to the boundary of the simplex in the internal metric tends to zero. Note that the extremality of the point is already implied by this condition. This property does not depend on the choice of the initial metric.

2. A central measure $\mu$ on the path space of a graph $\Gamma$ is called standard if there exists a sequence of vertices $\gamma_{n}\in\Gamma_{n}$ such that for every $\epsilon>0$ we have $E\lim\mu_{n}(V_{\epsilon}(\gamma_{n}))=1$, where $V_{\epsilon}(\cdot)$ is a neighborhood in the metric $\rho_{n}$. This property may be called the asymptotic concentration of measure on paths, or the law of large numbers; it does not depend on the choice of the initial metric.

3. A measure of maximal entropy (central measure) on a Markov compactum $\cal X$ is called standard if it has the following property: the distance in the metric $\rho_{n}$ between the conditional distribution on $X_{n}$ (given a fixed condition on $X_{n+1}$) tends to zero in measure. This property is a strengthening of the theorem on convergence of martingales, since the convergence of measures in the internal metric means the uniformness of the ordinary convergence in the dimension of finite-dimensional distributions; it does not depend on the choice of the initial metric.

Extreme points of a projective limit of simplices ( = ergodic central
measures) that are not standard are called nonstandard.

A projective limit of simplices (respectively, a graph, a Markov compactum) is
called standard if its all extreme points (ergodic measures) are standard.

In the theory of filtrations [11, 13], the notion of standardness was
introduced as a nonobvious generalization of the independence property; for
instance, a homogeneous ergodic filtration is the filtration of the
$\sigma$-fields of pasts of a sequence of i. i. d. variables (Bernoulli
scheme) if and only if it is standard (the standardness criterion from [11]).
In the general homogeneous, and even semi-homogeneous, case, there are many
nonisomorphic standard filtrations which are quite complicated. But a common
property, which makes them akin to independence, is that for all such
filtrations there is an “additional” basis, i. e., a well-structured, with
respect to the filtration, family of measurable sets generating the whole
$\sigma$-field. In the theory of stationary processes, the standardness of the
filtration of pasts is a strengthening of the regularity (or Kolmogorov)
property, i. e., the triviality of the intersection of the filtrations. It is
natural to compare it with Ornstein’s very weak Bernoulli property [7]; these
two properties are in general position, but akin in the terminology of
formulations, the difference being in the metrics used on the spaces of
conditional measures.

The standardness condition itself may also be called the concentration in the
internal metric, since the point is that the projections of the measure are
concentrated near some point of the boundary in this metric, or that the paths
of the graph for large $n$ lie in a small $\rho$-neighborhood of a single
vertex. This property is sometimes called “the existence of a limit shape,”
but the precise meaning of these words depends on the metric. Finally, this
property may, with good reason, be called the generalized law of large
numbers, which is especially clear from the second statement. However, one
should keep in mind the role of the metric in limit-shape-like theorems.

We have singled out the class of projective limits (graphs, Markov compacta)
for which all ergodic central measures are standard, and we arrive at the
following fundamental problem.

######  Problem 2.

Describe the standard projective limits of simplices (standard graphs,
standard Markov compacta), e.g., in terms of their definition via the matrices
$\\{M_{n}\\}$ or in other terms. The asymptotic nature of this problem (in the
sense that the answer does not change if we change finitely many initial
matrices) allows one to hope only for sufficient conditions for standardness.

Below we will see that a number of classical examples (the Pascal graph, the
Young graph, random walks on groups, etc.) turn out to be standard. At the
same time there exist nonstandard examples, and this is the general case. The
suggested procedure allows one not only to establish the standardness when it
holds, but also to obtain a parametrization of the ergodic measures.

######  Definition 4.

A sequence $x_{n}\in\Sigma_{n}$ of vertices of the simplex $\Sigma$ (or a
sequence $x_{n}\in\Gamma_{n}$ of vertices of the graph $\Gamma$) is called
regular if there exists a standard extreme point $x_{\infty}$ of the
projective limit (respectively, a standard central measure on the path state
of the graph) for which the sequence of projections to the simplices
$\Sigma_{n}$ converges with the sequence $x_{n}$ in the internal metric.

Regular sequences break into classes parametrized by all standard extreme
points. One can easily check that the following regularity criterion holds.

######  Proposition 5.

A sequence $\\{x_{n}\\}$ is regular if it is a Cauchy sequence in the internal
metric (see the definition at the end of Section 3.2).

###  4.2 Uniform compactness

We begin with a simple lemma (proved as early as in [13]).

######  Lemma 1.

If the number of vertices at the level $\Gamma_{n}$ of a graph $\Gamma$
(respectively, the number of points in the state space $X_{n}$ of a Markov
compactum) for all $n$ is bounded by a constant not depending on $n$, then the
graph (Markov compactum) is standard, i. e., all its central ergodic measures
are standard.

###### Proof.

Note that the mean distance between points with respect to the iterations of
the metric $\rho_{n}$ tends to zero, since the number of points does not
increase and for at least one pair of points at the next step the distance
will decrease with a constant factor; moreover, no distance can remain
constant by the ergodicity, and the convergence to zero is uniform over all
measures on the given compactum. ∎

It is interesting what is the minimum growth rate of the number of vertices
for which nonstandardness appears.

We will consider the internal metric on the levels $\Gamma_{n}$ of the graph,
or on the state sets $X_{n}$ of the Markov chain.

######  Definition 5.

We say that the family $\Gamma_{n}$ is uniformly compact in the internal
metric if for every $\epsilon>0$ the number of points in an $\epsilon$-net for
$\Gamma_{n}$ is uniformly bounded in $n$.

If this condition is satisfied for the sets $\Gamma_{n}$, i. e., for the
vertex sets of the simplices, then it also holds for the family of the
simplices themselves endowed with the internal metric.

######  Corollary 2.

If there exists a nonstandard extreme point, then the family of the vertex
sets is not uniformly compact.

Indeed, the uniform compactness would mean that for every $\epsilon>0$ there
is a uniformly bounded set of points of $\epsilon$-nets on all levels, and it
would follow by Lemma 1 that every extreme point is standard.

The internal metric can be extended by continuity to the set of standard
extreme points (standard ergodic measures).

######  Theorem 2.

If the family of simplices $\\{\Sigma_{n}\\}_{n}$ (equivalently, the family of
their vertices) is uniformly compact in the internal metric, then the
projective limit of these simplices (respectively, the Bratteli diagram, the
Markov compactum) is standard. In this case, the limit is a Bauer simplex (=
the Choquet boundary is closed), and the internal topology on it coincides
with the weak topology. The set of extreme points coincides with the set of
limits of regular sequences.

###### Proof.

By Corollary 2, it follows from the uniform compactness in the internal metric
that every extreme point is standard. But every standard extreme point is the
limit of a regular sequence of vertices. The completion of the space of
(classes of) sequences with respect to the internal metric thus coincides with
the set of all extreme points. Therefore, the Choquet boundary is compact.
This also implies the coincidence of the topologies, since the identity map of
the simplex endowed with the internal topology into the same simplex with the
weak topology is, obviously, continuous. ∎

Briefly speaking, the uniform compactness of a family of simplices in the
internal metric means the compactness of the set of extreme points and the
projective limit of simplices in the internal metric.

Note that the set of standard extreme points may be empty, but such a simplex
itself may be or not be a Bauer simplex (i. e., have a weakly compact Choquet
boundary). For instance, there are examples in which the Choquet boundary is
closed, and even consists of a single point, but the extreme point is not
standard and the simplices do not enjoy the required compactness in the
internal topology.

######  Conclusion 1.

We suggest a method for solving the problem of enumerating the central
measures, which consists in establishing the uniform compactness of the family
of the vertices of the simplices, or the vertices of the graph, or the states
of the Markov compactum, in the internal metric. If the uniform compactness
holds, then the completion of these spaces with respect to this metric is the
set of all ergodic invariant measures. In this case, all these measures are
standard and satisfy the concentration condition. An explicit parametrization
of the set of ergodic measures follows from the construction. This conclusion
remains true also for the problem of enumerating measures with given
cotransitions (i. e., quasi-invariant measures), that is, for describing
entrance and exit boundaries, Martin boundaries, etc.

##  5 Comments and examples

###  5.1 Dependence on the approximation and lacunary isomorphism

Recall that, by the lacunary isomorphism theorem (for its homogeneous case,
see [13]), every filtration is lacunary standard, i. e., contains a standard
subsequence. It follows that if we allow to pass from a given graph (Markov
compactum) to the graph (compactum) in which some (possibly, infinitely many)
levels are omitted and the multiplicities of the new edges take into account
the omitted vertices and edges, then a nonstandard central measure can become
standard. We will call this operation a rarefaction. It changes the internal
topology: in the rarefied graph the topology becomes weaker. Thus, in general,
the stock of standard extreme points increases. Hence a rarefaction can turn a
nonstandard graph into a standard one, if it makes all measures standard
simultaneously.111111It is not clear whether there exists a graph for which
such a rarefaction is not possible; it is most likely that this will be the
case for the universal or random graded graph. However, in practice this
operation is very inefficient, and obtaining a standard graph from a
nonstandard one at this price is not reasonable. For example, for filtrations
with positive entropy, the required rarefaction grows superexponentially.
Moreover, by the essence of the question, it is natural to associate a
parametrization of the central measures with the original graph rather than
with a rarefied one.

###  5.2 Examples of calculations

Clearly, the practical conclusion from the above considerations is that one
should calculate the internal metric on the levels and check whether the
uniform compactness holds. This is a purely combinatorial problem, sometimes
easy and sometimes not that easy. In the compact case, the completion with
respect to the internal metric provides an explicit parametrization of the
extreme points.

######  Theorem 3.

The Pascal graph of any dimension (${\mathbb{Z}}_{+}^{d+1}$) is standard. The
list of ergodic invariant measures is the list of Bernoulli measures
parametrized by the points of the unit $(d-1)$-dimensional simplex (for $d=2$,
this is de Finetti’s theorem). The Young graph (see Fig. 1) is also standard.

The Pascal graph of dimension $d$ is the lattice ${\mathbb{Z}}_{+}^{d}$ graded
by the sum of the coordinates; for the ordinary Pascal graph, $d=2$. The
calculation of the internal metric in this case is outlined in [14]. In this
case, it suffices to use manipulations with binomial or multinomial
coefficients. The proof of the standardness gives another proof of the
generalized de Finetti theorem. We leave detailed calculations to a detailed
paper about examples. A nontrivial fact here is the appearance of the root
norm for the Cartan subalgebra of $\widehat{sl}(d+1,\mathbb{R})$ as the norm
corresponding to the internal metric.

The standardness of the Young graph implicitly follows from Thoma’s theorem
(i.e., from the list of characters), but there is every reason to hope that
one will be able to check the compactness in the internal metric directly and
thus obtain the first purely combinatorial proof of Thoma’s theorem. Besides
formulas for the dimensions (e.g., the hook-length formula), an approximation
of the graphs themselves may be useful: the uniform compactness in this case
can be obtained from an approximation of the Young graph by multidimensional
Pascal graphs of growing dimension. The Thoma parameter ranges over the
simplex of two-sided nonnegative converging series with sum at most $1$.

It is most likely that the standardness holds for a much more general case,
which covers all the previous ones, namely, for the graphs that are the Hasse
diagrams of distributive lattices with finitely many generators. In
particular, the multidimensional Young graphs belong to this class. It is
interesting that candidates for the role of parameters in all these cases are
the independent frequencies of certain (minimal infinite) ideals, and the
generalized independence announced above is exactly a certain independence of
these frequencies. The standardness (i. e., the compactness of the set of
standard extreme points) implies the compactness of the set of these
frequencies. The reason behind the uniform compactness in all these cases (see
above) lies in the following structure of the graphs: all vertices preceding
any given vertex are close in the internal metric, and this closeness
increases with the number of the level containing this vertex.

It is exactly the absence of this property that is a characteristic feature of
the known examples of nonstandard graphs. For the graph of unordered pairs
(see Fig. 2), one does not know the answers to simple questions related to the
internal metric, for instance, what is the growth rate of the
$\epsilon$-entropy. All these questions are closely related to the theory of
so-called tower of measures [13].

Translated by N. V. Tsilevich.

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# Polarization Spectroscopy of High-Order Harmonic Generation in Gallium
Arsenide

Shatha Kaassamani1  Thierry Auguste1  Nicolas Tancogne-Dejean2  Xu Liu1,3
Willem Boutu1  Hamed Merdji1  and David Gauthier1  1Université Paris-Saclay,
CEA, CNRS, LIDYL, 91191 Gif-sur-Yvette, France
2Max Planck Institute for the Structure and Dynamics of Matter and Center for
Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
3Imagine Optic, 18, rue Charles de Gaulle, 91400 ORSAY, France
<EMAIL_ADDRESS>
###### Abstract

An interesting property of high harmonic generation in solids is its laser
polarization dependent nature which in turn provides information about the
crystal and band structure of the generation medium. Here we report on the
linear polarization dependence of high-order harmonic generation from a
gallium arsenide crystal. Interestingly, we observe a significant evolution of
the anisotropic response of above bandgap harmonics as a function of the laser
intensity. We attribute this change to fundamental microscopic effects of the
emission process comprising a competition between intraband and interband
dynamics. This intensity dependence of the anisotropic nature of the
generation process offers the possibility to drive and control the electron
current along preferred directions of the crystal, and could serve as a
switching technique in an integrated all-solid-state petahertz optoelectronic
device.

††journal: osajournal††articletype: Research Article

##  1 Introduction

Since its first observation in 2010 [1], continuous and extensive research has
been carried out to understand the fundamental physical processes underlying
high harmonic generation (HHG) in semiconductors and dielectrics. HHG was
investigated in a number of crystals including semiconductors and insulators
[1, 2, 3, 4], as well as 2D materials [5, 6, 7, 8]. An interesting
characteristic of HHG in crystals is the strong dependence of the generation
process on the structural and optical properties of the medium, and its
response to the laser parameters, such as intensity and polarization. This
dependence has been exploited as a spectroscopic tool to access, for instance,
the electron dynamics and retrieve the band structure [9, 10, 11, 12, 13, 14,
15, 16, 17]. The polarization-dependent nature of the HHG process has been
explained from a real-space picture [2, 18] as a result of the crystal lattice
structure, or correspondingly from a momentum-space domain representation
which involves the crystal band structure [9, 10]. As a direct consequence of
the band configurations in momentum space, two main mechanisms are involved in
the emission of photons for a given harmonic: the interband polarization and
the nonlinear intraband current. The former necessitates the recombination of
an electron–hole pair between two energy bands, while the latter originates
from the nonlinear oscillation of electrons in non-parabolic band curvatures.
These two mechanisms yield a generation efficiency that reflects the crystal
or band structure. For instance, Lanin et al. [19] have mapped the anisotropic
band dispersion of a ZnSe crystal through the polarization and intensity-
dependent below bandgap harmonics, dominated by intraband currents. However,
for above bandgap harmonics both interband and intraband mechanisms can
contribute to the harmonic emission, and a competition might arise between
them especially when electrons explore higher energy regions of the band
structure. Here, we report on our study of high harmonic generation in gallium
arsenide (GaAs) both in transmission and reflection geometries. We investigate
the dependence of above bandgap harmonics on the laser polarization for
different laser intensities in the nonperturbative regime. We find out that
the polarization dependence of low order harmonics changes with increasing
laser intensity, in such a way that the maximum harmonic yield reverses from
one crystal axis to another. In the transmission geometry, we characterize the
laser propagation through the crystal and prove that nonlinear propagation
effects have minimal impact on the change in the HHG anisotropy maps measured.
This is further supported by measurements carried out in reflection geometry,
where the same intensity-dependent polarization mapping is obtained. We
conclude that our observations portray pure electron dynamics involved in
interband and intraband mechanisms in GaAs.

##  2 Experimental Setup and Details

The HHG experiment in transmission geometry (t-HHG) is performed by employing
an all-fiber mid-IR laser (NOVAE Brevity [20]), delivering 82 fs, 9 nJ laser
pulses at a repetition rate of 18.66 MHz centered at a wavelength of 2.1 µm.
The beam is focused by a 3 cm focal length lens onto a 500 µm thick gallium
arsenide (GaAs) crystal having a (100) cut. The laser intensity and
polarization are varied by a half-wave plate and a polarizer. To take into
account propagation effects in the GaAs sample, the output power as well as
the beam size near the exit surface are measured to estimate the effective
intensities responsible for the HHG. This is detailed in section 3.2. For the
HHG experiment in reflection geometry (r-HHG), we employ an OPCPA (Optical
Parametric Chirped Pulse Amplification, Starzz from Fastlite) laser system
providing higher energy pulses, reaching 10 µJ, at a central wavelength of 2.4
µm. The repetition rate is 100 kHz and the pulse duration is 75 fs. The laser
pulses are focused by a 7.5 cm focal length lens at ${45^{\circ}}$ angle of
incidence onto the 500 µm thick GaAs (100) crystal. The estimated beam size at
the focus on the sample is around 40 µm in the vertical direction. The
reported effective intensities in the reflection configuration are estimated
taking into account the incidence angle and the Fresnel transmission
coefficient for an s-polarized state. For both experiments, t-HHG and r-HHG,
the harmonics generated are collected and focused by a 10 cm focal length lens
onto a CCD camera and a spectrometer. The spectral selection of each harmonic
is done using band pass filters centered at each harmonic wavelength. Given
that GaAs is a narrow direct bandgap semiconductor with a bandgap energy of
1.4 eV, all generated harmonics are above bandgap harmonics. The direct
bandgap emissions are measured in the spectrum from GaAs around a wavelength
of 880 nm. In addition, another less intense fluorescence signal at 450 nm is
measured, which corresponds to higher direct transitions in the conduction
bands at the ${\Gamma}$ point. These two fluorescent signals are filtered out
spectrally or discriminated spatially on the camera.

##  3 Results

###  3.1 HHG in Transmission Geometry

A schematic representation of the geometry of the t-HHG experiment, including
the relative orientations of the crystal axes with the laser polarization, is
shown in figure 1(a). The laser polarization is initially aligned along the
${\Gamma K}$ direction of the crystal in its (100) plane (i.e. at
$\mathrm{\theta=0^{\circ}}$). Being limited to the maximum pulse energy
delivered by the laser, harmonics up to the $\mathrm{7^{th}}$ order (300 nm)
are efficiently generated in GaAs.

Figure 1: (a) Schematic representation of the HHG experiment in transmission
(t-HHG). (b) The third (H3), fifth (H5) and seventh (H7) harmonic signals
emitted along the ${\Gamma K}$ and ${\Gamma X}$ axes of the GaAs crystal as a
function of the effective laser intensity ${\it{I_{eff}}}$ calculated in
section 3.2. The perturbative scaling is plotted in a dashed black line. (c)
The polarization dependences of harmonics 3, 5 and 7 of the 2.1 µm driving
laser generated from GaAs at different laser intensities. The indicated
intensities are the estimated ones along the ${\Gamma K}$ direction. Note that
the normalization factor of each scan is chosen only for a good visualization
of the polarization dependences.

The dependence of each harmonic yield on the laser intensity is measured along
the two main crystal axes ${\Gamma K}$ and ${\Gamma X}$, and shown in a double
logarithmic scale in figure 1(b). The scaling of harmonics 3 (H3), 5 (H5) and
7 (H7) clearly drifts from the perturbative scaling law (represented in black
dashed lines), which verifies that the HHG process takes place in the non-
perturbative regime. In addition, a remarkable behavior in the laser intensity
dependence of the $\mathrm{3^{rd}}$ and $\mathrm{5^{th}}$ harmonic emissions
is observed. As can be seen in figure 1(b), at low intensities the harmonic
generation is more efficient along the ${\Gamma K}$ direction of the GaAs
crystal (blue curves). However, at an intensity around 0.15
$\mathrm{TW/cm^{2}}$, an inversion in the yield of H3 and H5 between the two
main axes takes place, in which the $\mathrm{3^{rd}}$ and $\mathrm{5^{th}}$
harmonic emissions become stronger along the ${\Gamma X}$ direction. To push
these findings further, we carry out polarization-dependent measurements of
each generated harmonic at different laser intensities. The corresponding
polar plots are depicted in figure 1(c). The laser polarization is initially
aligned along the ${\Gamma K}$ direction of the crystal, and the polarization
is then rotated in its (100) plane. At a low laser intensity (0.07
$\mathrm{TW/cm^{2}}$ \- violet plots), the polarization dependence of all
harmonics reveals a similar fourfold symmetry. In this case, the harmonic
efficiency is maximum when the laser polarization is aligned along the
${\Gamma K}$ direction of the crystal (at $\mathrm{0^{\circ}}$ and by every
$\mathrm{90^{\circ}}$), and is minimum along the ${\Gamma X}$ direction
($\mathrm{45^{\circ}}$ rotation from the ${\Gamma K}$ directions). Such
anisotropy is expected as GaAs has a zincblende crystal structure and thus
reveals the fourfold symmetric nature of the crystal. As the harmonic order
increases, for instance for H7, a stronger dependence is measured.
Surprisingly, for slightly higher laser intensities, a change in the
polarization dependences of the $\mathrm{3^{rd}}$ and $\mathrm{5^{th}}$
harmonic emissions is observed. For H3, as we increase the laser intensity the
contrast of the anisotropic response diminishes, while for H5, its generation
efficiency becomes isotropic at an effective intensity of 0.13
$\mathrm{TW/cm^{2}}$. For higher laser intensities, a rotation by
$\mathrm{45^{\circ}}$ in the polarization dependence is observed, where the
harmonic signal maximizes along the ${\Gamma X}$ crystal axis instead. This
change and dependence become stronger at our maximum laser intensity of 0.29
$\mathrm{TW/cm^{2}}$. On the other hand, the polarization dependence of H7 at
any laser intensity remains the same.

###  3.2 Laser Propagation Effects

Given the solid state nature of the generation medium, nonlinear propagation
effects can come into play for thick crystals (few tens of micrometers). This
can significantly impact the laser propagation through the crystal before
reaching the effective harmonic generation region [21, 22, 23]. Therefore, it
is necessary to disentangle these effects from the real physical mechanism of
high harmonic generation. GaAs is characterised by its high linear and
nonlinear refractive indices at wavelengths around 2.1 µm, where ${n_{0}}$ =
3.38 and ${n_{2}}$ = $\mathrm{10^{-13}\,cm^{2}/W}$ [24, 25], respectively,
almost three orders of magnitude higher than those of $\mathrm{SiO_{2}}$ [26].
Besides, the nonlinear refractive index and the multi-photon absorption
coefficient at a wavelength of around 2 µm are slightly larger along the
${\Gamma K}$ than along the ${\Gamma X}$ axis of the crystal. Consequently,
the Kerr effect is expected to be stronger along the ${\Gamma K}$ direction.
Indeed, Xia et al. [21] previously showed that the anisotropic multiphoton
absorption in GaAs results in an increase of the harmonic emission along the
${\Gamma K}$ direction of a thick GaAs crystal compared to a thin one. In our
case, the absorption of the fundamental beam along the two axes of the crystal
is estimated by measuring the transmitted output average power after the GaAs
sample. This includes the Fresnel transmission and the nonlinear absorption in
the crystal. The measurements for different laser input intensities (as
calculated from the input power and the focal spot without the sample) are
reported in figure 2(a).

Figure 2: (a) The ratio of the output power to the input power of the laser,
(b) the estimated beam size at full width half maximum, and (c) the calculated
effective intensity ${\it{I_{eff}}}$ close to the rear surface of the GaAs
crystal versus the input laser intensity ${\it{I_{in}}}$ along the two main
crystal axes ${\Gamma K}$ (in blue) and ${\Gamma X}$ (in red).

In agreement with previous studies, the absorption along the ${\Gamma K}$
direction of GaAs is slightly stronger than that along ${\Gamma X}$ axis. In
addition, we characterize the self-focusing effect by calculating the
fundamental mode size at the exit of the GaAs crystal from the direct
measurement of the $\mathrm{5^{th}}$ harmonic mode. The results are displayed
in figure 2(b). Our measurements reveal that with increasing laser intensity
the fundamental beam size decreases, and at a faster rate when the
polarization is along the ${\Gamma K}$ direction than along ${\Gamma X}$.
Finally, combining the output power measurements and the beam size results in
a slightly higher effective intensity along the ${\Gamma K}$ axis (figure
2(c)). Consequently, this would not explain the pronounced $\mathrm{3^{rd}}$
and $\mathrm{5^{th}}$ harmonic yields along the ${\Gamma X}$ direction with
the increase of the laser intensity. Moreover, it is worth noting that in our
tight focusing geometry, since the Rayleigh length is much shorter than the
sample thickness, the overall nonlinear propagation effects are minimized. A
numerical study of the beam propagation shows that it is mainly ruled by
ionization-induced refraction and absorption by the plasma produced on the
leading front of the pulse, on the one hand, and by the Kerr-induced self-
focusing, on the other hand. For our laser parameters, the latter
counterbalances plasma defocusing and leads to the reduction of the beam size
observed in the experiment. In the temporal and spectral domains, a
modification of the laser pulse and a spectral blue shift take place, mainly
governed by photoionization of the valence band of the generation medium [27].
However, no clear difference was found between propagation along the ${\Gamma
K}$ and ${\Gamma X}$.

###  3.3 HHG in Reflection Geometry

To confirm the change in the polarization dependence of low order harmonics
generated in GaAs, we explore the polarization mapping of each individual
harmonic generated in reflection at different laser intensities. This geometry
excludes the nonlinear propagation effects that could act on the laser beam
and alter the pure microscopic interaction with the generation medium [28].
The geometry of the r-HHG experiment is schematized in figure 3(a). The laser
is set for s-polarization, and in this case the sample is rotated about the
normal to its (100) plane during the measurements. With the higher intensities
provided by the OPCPA laser system, high-order harmonics are efficiently
generated from GaAs, limited by our detection system to the $\mathrm{11^{th}}$
order. Damage on the sample appears above an effective intensity of 0.35
$\mathrm{TW/cm^{2}}$. The intensity scaling of harmonics 3 (H3), 5 (H5) and 7
(H7) are extracted from their polarization dependence measurements at
different laser intensities (figure 3(b)). Interestingly, the inversion in the
$\mathrm{5^{th}}$ harmonic yield appears again starting at a laser intensity
of around 0.15 $\mathrm{TW/cm^{2}}$. Figure 3(c) reports the polarization
mapping of the generated odd-order harmonics.

Figure 3: (a) Schematic representation of the HHG experiment in reflection
geometry (r-HHG). (b) The third (H3), fifth (H5) and seventh (H7) harmonic
signals emitted along the ${\Gamma K}$ and ${\Gamma X}$ axes of the GaAs
crystal as a function of the effective laser intensity ${\it{I_{eff}}}$. The
perturbative scaling is shown by the dashed black line. (c) The polarization
dependences of the harmonics H3-H11 of the 2.4 µm driving laser generated from
GaAs at different laser intensities.

Even though the wavelength used in reflection is different, similar results as
those of the t-HHG experiment are obtained. Particularly, the remarkable
change in the polarization dependence of the $\mathrm{3^{rd}}$ and
$\mathrm{5^{th}}$ harmonic emissions as a function of the laser intensity is
observed. At a low intensity of 0.07 $\mathrm{TW/cm^{2}}$, H5 signal is
maximum when the laser is aligned along the ${\Gamma K}$ direction, while for
relatively higher laser intensities, starting at 0.17 $\mathrm{TW/cm^{2}}$ and
above, the polarization dependence of H5 rotates by ${45^{\circ}}$, i.e. the
harmonic signal maximizes when the polarization is along the ${\Gamma X}$
direction. For higher order harmonics, H7, H9 and H11, the dependence on the
laser polarization remains the same for different intensities. Therefore, the
results of the polarization dependence of the HHG process in GaAs in
reflection corroborate those obtained in transmission, and strongly confirm
that the variation in the HHG anisotropy is mainly determined by the
microscopic response of the crystal.

##  4 Discussions

The intensity-dependent polarization mapping observed for low-order harmonics,
H3 and H5, generated from GaAs could originate from its band structure, which
impacts the emission mechanisms related to the intraband and interband
dynamics. A basic calculation of the electron excursion in the conduction
band, using the acceleration theorem, shows that the electron can explore
about ${25\%}$ of the Brillouin zone when driven at a laser intensity of 0.29
$\mathrm{TW/cm^{2}}$. Around this intensity, the electron wave packet
therefore reaches a region in the band structure where a small gap exists
between the first and second conduction bands of GaAs along the ${\Gamma X}$
direction [29], as shown in figure 4.

Figure 4: Band structure of GaAs along the main crystal axes of interest,
${\Gamma K}$ and ${\Gamma X}$, obtained by density-functional theory (DFT)
calculations [30, 31]. The gap opening in the conduction bands’ region induces
a significant inflection of the first conduction band (highlighted in red),
which is the source of intraband emission. Interband transitions, between the
light hole and first conduction band, corresponding to the harmonic energies
of the 2.1 µm driving laser are indicated.

Around such gap, Bloch oscillations are expected to be pronounced, leading to
significant nonlinear intraband current and consequently an enhancement in the
low order harmonic emission [1, 32]. This stronger contribution of the
intraband current with the increase of the laser intensity can explain the
rise of the $\mathrm{3^{rd}}$ and $\mathrm{5^{th}}$ harmonic yields along the
${\Gamma X}$ direction of GaAs. However, for higher order harmonics such as
harmonics 7, 9 and 11, the generation is expected to remain dominated mainly
by interband transitions, being more efficient along the ${\Gamma K}$
direction [33, 34]. Notice that the interband transitions for H3, H5 and H7
are allowed energetically and are relatively close to each other along the
${\Gamma X}$ direction. We would then expect that their yields evolve
similarly for a dominant interband transition with the laser intensity, which
is not what we observe. To support our hypothesis, we compare the
polarization-dependent HHG emission from GaAs to silicon, given that both
semiconductors possess the same lattice system, as well as similar valence and
first conduction bands along the ${\Gamma K}$ direction, although the direct
bandgap of silicon is larger than that of GaAs [29]. However, the small
additional gap present between the first and second conduction bands of GaAs
along the ${\Gamma X}$ direction does not exist in silicon. Consequently, all
harmonic orders present similar polarization dependences at high laser
intensity, as shown in figure 5.

Figure 5: (a) Schematic representation of the HHG experiment in a 1 µm thin
silicon (100) membrane. (b) The polarization dependences of harmonics 3 (H3),
5 (H5) and 7 (H7) of the 2.1 µm driving laser generated from Si at an
estimated effective laser intensity of 0.2 $\mathrm{TW/cm^{2}}$.

##  5 Conclusion

In summary, we have investigated high harmonic generation from GaAs in the
ultraviolet and visible spectral range, both in transmission and reflection
geometries. An interesting intensity-dependent polarization mapping of low
order harmonics is observed. We have successfully shown that nonlinear
propagation effects are negligible in comparison to the pure HHG response, and
that the change in the polarization dependence of the harmonics with the laser
intensity originates from microscopic mechanisms of the generation process.
The results of our work point out an interesting competition between the
intraband and interband electron dynamics in the conduction band of GaAs in
the strong-field regime. This would contribute to the fundamental
understanding of the HHG process in semiconductors, and provide a means to
differentiate between the two mechanisms involved. Finally, solid-state HHG
could mark an exciting development of an all-optical band structure
reconstruction technique.

## Funding

We acknowledge the financial support from the French ASTRE program through the
“NanoLight” grant, the ANR (PACHA grant number ANR-17_CE30-0008-01), as well
as the support from the PETACom project (Petahertz Optoelectronics
Communication) FET Open H2020 grant number 829153, OPTOLogic project (Optical
Topologic Logic) FET Open H2020 grant number 899794, and TSAR project
(Topological Solitons in Antiferroics) FET Open H2020 grant number 964931.

## Acknowledgments

The authors would like to truly thank Laure Lavoute, Dmitry Gaponov, Nicolas
Ducros and Sébastien Février for providing the NOVAE Brevity laser.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

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