Cybernetics and Systems Analysis, Vol. 40, No. 4, 2004

ERGODIC THEORY AND ARITHMETICAL

SIMULATION OF RANDOM PROCESSES

N. M. Glazunov,a

L. P. Postnikova,b

and N. Z. Shora UDC 519.6:519.21:511

The relationships between arithmetical simulation of random processes, ergodic theory, and

optimization are analyzed. Some new results are considered and their possible applications to

optimization problems are described.

Keywords: normal sequence, equidistribution, ergodic transformation, optimization.

Direct methods of stochastic optimization are based on stochastic approximation of random processes. The

corresponding calculations require generation of pseudorandom sequences that satisfy certain properties of probabilistic

distributions. Since stochastic optimization remains to be one of the fundamental research area at the Institute of Cybernetics,

National Academy of Science of Ukraine, the authors believe that it is expedient to consider some fundamental results of

stochastic approximation, in particular, to draw attention to economy methods of generating pseudorandom sequences based

on arithmetical principles [1]. The purpose of this paper is to review state-of-the-art of arithmetical methods of generating

such sequences and their ergodic properties. The review is based on studies by A. G. Postnikov, A. N. Kolmogorov,

N. M. Korobov, and their colleagues. Some applications and current state of the fundamental concepts they have developed

are described. Yu. M. Ermolyev, V. I. Norkin et al. develop direct methods of stochastic approximation at the Institute of

Cybernetics. One of the supervisors of stochastic optimization research was Director of the Institute, Academician V. S.

Mikhalevich. As a postgraduate student, he studied under the guidance of the Academician A. N. Kolmogorov at the

Moscow State University in 1953–1956 [2]. The supervisor awoke Mikhalevich’s interest in arithmetical nature of the

probability problems he studied, which resulted in collaboration with A. G. Postnikov and then with his colleagues [1, 3, 4].

The monograph “Arithmetical Simulation of Random Processes” by A. G. Postnikov [1] was published in 1960. The author

summarized intensive studies conducted since the beginning of the last century to establish arithmetical grounds for the

concepts of randomness and random sequences from the viewpoint of their uniform distribution [5–10]. New finding in this

subject area were actively discussed in the scientific world. A. N. Kolmogorov was interested in arithmetical simulation of

random processes, which favored discussions of the results on the subject in the Moscow Mathematical Society, at seminars

at the Moscow State University, and at the V. A. Steklov Institute of Mathematics, and then at conferences [1, p. 8]. A

sequence is uniformly distributed if the measure (probability measure if the measure of the whole domain is equal to unity)

of the number of times members of this sequence fall into a given subregion is proportional to Euclidean size of this

subregion. There are sequences for which the frequency its members fall into a given domain is not proportional to the size

of this domain, but there exists a measure distribution density characterizing the frequency of finding members of this

sequence in various places of the main domain. Such sequences are arithmetical analogs of random (pseudorandom)

sequences. A sequence is said to be constructed arithmetically if it is defined by a primitively recursive function [1]. The

“randomness” of a sequence is interpreted through some arithmetical properties of its subsequences. Let us consider

sequences that fill, uniformly or uniformly with some density, a domain or its subdomain of fixed dimension, and touch on

some modern aspects of the development and application of this subject such as ergodic theory and optimization on

Riemannian surfaces.

5271060-0396/04/4004-0527©

2004 Springer Science+Business Media, Inc.

aV. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kiev, Ukraine,glanm@d105.icyb.kiev.ua; nzshor@d120.icyb.kiev.ua. bMoscow Pedagogical University, Moscow, Russia. Translated fromKibernetika i Sistemnyi Analiz, No. 4, pp. 73-86, July-August 2004. Original article submitted November 19, 2003.

THE CONCEPT OF RANDOMNESS AND NORMAL SEQUENCES

Let us follow [11–14] presented in [1, 15]. In [11], I. Venn outlined history of the problem and pointed out that

randomness should be determined in terms of frequencies. R. Mises introduced the term “collective” in [12]. A collective is

determined by two requirements:

1) relative frequencies of different indications have definite limiting values;

2) the limiting value of the indication frequency remains constant if any part is arbitrarily selected from the whole

sequence and only this part is considered hereafter (irregularity of the collective).

Along with the analysis of logical and arithmetical foundation of the concept of randomness, various “tables of

random numbers” or “tables of pseudorandom numbers” were compiled to meet practical needs. Idealization of such tables

are tables not bounded in one leg or the tables not bounded in both legs, i.e., in any case, we deal with an infinite sequence of

numbers.

In 1914, E. Borel introduced the concept of a weakly normal number [6, 1]: a real number � �, 0 1� � , is called

weakly normal with respect to the basis g if each sign of the sequence � �1 2, , . . . , obtained as a result of decomposition of �

into an infinite g-nary fraction � � �� � �1 22/ /g g �3

3/ . . .g � , appears with asymptotic frequency equal to 1/ g.

Further, E. Borel calls the number � absolutely normal if it is weakly normal with respect to any natural basis g greater than

1. Based on the theory of measure, he established that absolutely normal numbers exist. W. Sierpinski [9] and H. Lebesgue

[7] analyzed absolutely normal numbers. Let us recall the concept of a normal sequence of signs [1]. Let

� �1 2, , . . . , (1)

be an infinite sequence consisting of signs 0 1 1, , . . . , g � . Let us select natural s and write the sequence of s-term

brackets:

( , , . . . , ) ( , , . . . , ) ( , , . . . , )� � � � � � � � �1 2 2 3 1 3 4 2s s s� � ... (2)

Sequence (2) is sometimes called [1] a caterpillar (of rank s) of sequence (1). Let � be a fixed s-term bracket

consisting of signs 0 1 1, , . . . , g � . Denote by Wp ( )� how many times the bracket � appears up to the pth term of sequence

(3). The sequence of signs (1) is called a normal sequence of signs if

lim ( ) / /p

psW p g

���� 1

for any natural s and any s-term bracket.

Example 1 (D. Champernowne [16]). Let us construct, following [1, 16] , an example of a normal sequence of signs.

Let S r be the sequence of all r-digit numbers in the g-nary numerical system, and a combination of signs that begin from

zero, which is also considered an r-digit number. The numbers are taken in their natural order. For example, for g � 2 ,

S

S

S

1

2

3

0 1

00 01 10 11

000 001 010 011 100 101 110

� �

� � � �

� � � � � � � �111

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Writing S r , we put commas from above between the r-digit numbers. Champernowne proves that the sequence

S S S S r1 2 3, , , . . . , , . . . , (3)

(since S r is a group of signs but not one sign, the notation is symbolic) is normal, i.e., in the caterpillar of sequence

(3), each �-digit combination of signs (natural � is fixed) � � appears with asymptotic frequency1

g �.

Let g 2 be a natural number and � be real, 0 1� �� . Let us consider a sequence of fractional parts { }�g x , where

x �1 2, , . . . . Denote by � an interval on the interval [ , ]0 1 , | |� being its length and Wp ( )� the number of fractional parts { }�g x ,

x p�1 2, , . . . , , fallen into the interval �. According to H. Wale [5], the sequence of fractional parts { }�g x is uniformly

528

distributed over the interval [ , ]0 1 if

lim( )

| |p

pW

p���

��

holds for any interval �.

Let us decompose � into an infinite gth fraction:

� � � �� � � �1 22

33/ / / . . .g g g (4)

According to the theorem from [14], a uniform distribution of fractional parts { }�g x , x �1 2, , . . . , on the interval [ , ]0 1

is equivalent to the fact that the sequence � �1 2, , . . . is normal. This is one more example of a normal sequence of signs.

Let us recall the concept of a uniformly distributed sequence in terms of distribution function, slightly modifying the

definition from [17]. Let us formulate it for a compact interval of a real axis. Let � � [ , ]a b be a subinterval of the interval

, and # ( )� �n be the number of terms of the sequence � � �� ( , . . . , , . . . )1 j such that j n� and a bj� �� .

The function Vn

nn( )

# ( )�

� �� is called a distribution function for the first n terms of the sequence � on the interval .

If the limit lim ( ) ( )n

nV V��

�� � exists for any �, then V ( )� is called the distribution function for the sequence �.

Note that V ( )0 0� and V ( ) �1. The case where V ( )� �� is an important special case. This corresponds to a uniform

(according to H. Wale) distribution.

Recall that problems of distribution of fractional parts of an exponential function are sometimes subdivided into two

groups:

1) an individual theory, when distribution of fractional parts of the exponential function { }�g x is considered for the

given fixed number � ;

2) a metric theory, in which the measure of the set of numbers �, for which a property of a sequence of fractional parts

of an exponential function is true, is analyzed.

According to the H. Wale theorem [3], the sequence of fractional parts { }�g x , x �1 2, , . . . , is uniformly distributed

over the interval [ , ]0 1 for almost all (on the Lebesgue measure) numbers � �, 0 1� � . This theorem also proves that normal

sequences of signs exist.

A. G. Postnikov and I. I. Pyatetski–Shapiro [9] introduced the concept of a normal (according to Bernoulli) sequence of

signs. Let two positive numbers p and q be given such that p q� �1, and there be an infinite sequence consisting of 0 and 1,

� � �1 2 3, , , . . . (5)

Let s be any natural number. We write (5) as a “caterpillar”:

( , , . . . , ) ( , , . . . , ) . . . ( , . . . ,,� � � � � � � � �1 2 2 3 1 1s s p p p s� � � �1 ) . . . (6)

Let � � ( , . . . , )� �1 s be any s-term bracket consisting of 0 and 1, and Wp ( )� show how many times the bracket �appears up to the pth term of sequence (6). Sequence (6) is called normal (according to Bernoulli) sequence of signs if

lim ( ) /p

pj s jW p p q

��

���

for any natural s and any s-term bracket, where j is the number of unities among the signs � � ( , . . . , )� �1 s .

The strong law of large numbers for stationary random sequences [18, 417] leads to the theorem.

THEOREM A. Let an unlimited number of independent tests be performed. In each test, an event denoted by 1 has

probability p, and the event denoted by 0 has probability q. With probability 1, the sequence of results is normal according to

Bernoulli.

As A. G. Postnikov mentioned, existence of normal (according to Bernoulli) sequences of signs follows from this

theorem. The following fact justifies the concept of a normal (according to Bernoulli) sequence of signs introduced to clarify

the term “table of pseudorandom numbers.” As A. G. Postnikov showed [19], such or more general sequences of numbers

are apparently enough to construct a numerical method similar to the Monte Carlo method but with a reliable error estimate.

Other ways of constructing normal sequences of signs are given in by A. Copeland and P. Erdos [20] and G. Devenport and

Ð. Erdos [21]. A. G. Postnikov pointed out that the following criterion (established by I. Pyatetskii-Shapiro [22]) simplifies

proof of the fact that the sequence written using the Champernowne mode is a normal sequence of signs.

529

THEOREM B (I. Pyatetskii–Shapiro). Let the sequence

� � �1 2 3, , , . . . , (7)

consisting of the signs 0 1 1, , . . . , g � , be such that there is a constant C � 0 such that

lim( )

R

R

s

N

R

C

g���

�

for any natural s and any s-term combination. Then sequence (7) is a normal sequence of signs.

A. G. Postnikov and I. Pyatetskii-Shapiro [3] extended the Champernowne’s method and constructed a normal

(according to Bernoulli) sequence of signs for arbitrary p. A theorem similar to the above-mentioned criterion is used in this

case. In a similar way, A. Postnikov and I. Pyatetskii-Shapiro gave normal implementation of the elementary stationary

Markov chain and normal implementation of the process corresponding to a continued fraction [3].

SOME RANDOM PROCESSES AND THEIR GENERATION

In many studies carried out at the Institute of Cybernetics, in particular, by Yu. M. Ermol’ev and his colleagues [23,

24] and his followers P. S. Knopov et al., employ pseudorandom and random sequences (see below). One of the methods for

representing random sequences is the method of their arithmetical generation. A. G. Postnikov and N. M. Korobov are one of

the initiators of the method of arithmetical simulation of random sequences. L. P. Postnikova develop some of the

above-mentioned methods [4, 25]. Arithmetical-algebraic methods of generation of such sequences are quite efficient since

they require realization of a relatively small number of arithmetical operations. V. S. Mikhalevich, being Director of the

Institute of Cybernetics, promoted the development of the methods of stochastic simulation based on stochastic

approximation in solving respective optimization problems.

Let us consider probability-theoretic and arithmetical approaches to analysis of random processes. The measure of

some set is generalization of the length of an interval, area of a geometrical figure, or volume of a body. Intuitively, it

corresponds to the mass of a set from some space for given mass distribution in this space. More precisely, let X be a set and

be a class of its subsets. A non-negative function of the set �, defined on , is called additive, finite-additive, or

countable-additive if

� ��i

n

i

i

n

iU U

� �

�

�

��

�

�

��� �

1 1

( )

for any Ui � ,

i

n

iU

�

�1

� , U Ui j� � 0 , i j� , for n � 2, a finite n, and n � �, respectively. In axiomatics of probability

theory, which we use [26] and consider known, the concept of probability space ( , , )M � is introduced, where M is

the space (called sometimes the space of simple events), � � is algebra of subsets from M, and � is

countable-additive measure such that � ( )M �1 (a probability measure). The countable additivity in this definition cannot

be replaced with finite additivity. Following [27], let us present an example of a finite-additive function, which satisfies

all the axioms of a measure, except for countable additivity. Let M be a subset of the set of natural numbers.

A logarithmic density can be defined as a function of the form

ld ( ) limln ,

MN nN

n N n M

��� � �

�1 1.

(8)

If the limit in (8) does not exist, then the set M is called nonmeasurable with respect to logarithmic density. Yu. I.

Manin constructed an example of the above-mentioned function, not being countable-additive (for details see [27, pp. 187,

188]. Let us modify the example: let the set of natural numbers be divided into classes Ai , where each class is an arithmetical

progression of the form

A l P x P xi i i i� � �{ }, , , , . . .0 1 , li is natural,

and the terms li are selected as follows: let Pi be a prime number greater than 2, P Pii�

1, then l l1 21� , is equal to the

least natural number greater than unity and not lying in A1, l3 is equal to the least natural number, which is not lying

530

in the union A A1 2� , etc. Then classes Ai do not intersect (A Ai j� � 0 for i j� ),

i

n

iA� � N. In this case,

ld ( )AP

ii

�1

and

i iP� 1

, while ld ( )N �1, i.e., countable additivity does not take place. For example, for P1 5� ,

P iii� �5 1 2, , ,... , A1 1 6 11� , , , ... , A2 2 27 52� , , , . . . , A3 3 128 253� , , , . . . ,

ii� �

1

5

1

4. If we consider

1

Pi

as the

probability of selecting ith progression of Ai (a simple event) from the set N (space of simple events), then due to

i iP� �

11 we arrive at inconsistency with the property of a probability measure. It is possible to prove the following.

Proposition 1. The set of sequential prime numbers beginning from number 3 parametrizes the set of not

countable-additive functions defined by classes Ai .

We believe further that the property of countable additivity holds .

In terms of the theory of functions, a random process is a real function � ( , ), which is a measurable function of

�M for each fixed �T, where T is the set of values of the parameter [28]. It is convenient to use the equivalent

definition of a random process using space of infinite sequences [1, 29, 30]. We restrict ourselves to discrete random

processes. Let us first recall, following P. Halmosh [29] and A. G. Postnikov [1], interpretation of a discrete random process,

space M is space of infinite sequences, and a measure on M is determined using a measure on elementarily cylindrical

subsets of the set M. Let us present only exact definitions, with a slightly changed notation. Let g 2 be a natural number,

X g� �{ }1 2 1, , . . . , be the alphabet of signs 1 2 1, , . . . , g � , and X be the set composed from the signs 1 2 1, , . . . , g � of all

infinite sequences (space of superwords in the alphabet X ). Let r be any natural number. By an elementary-cylindrical (e.c)

set of rank r is meant a set of sequences of the form

a a X a Xr i1 , . . . , , � .

Elementary-cylindrical sets include also the set X and an empty set. By a cylindrical (c.) set is meant a set, which

may be represented as a finite union of e.c.-sets. Let us recall the well-known facts about c.-sets [1, 29] .

Proposition 2. Intersection of two c.-sets is a c.-set. An addition for a c.-set is a c.-set. Thus, c.-sets form an algebra.

The measure � is determined on the class of e.c.-sets as follows: � ( )X �1, � ( )� � 0, where � is an empty set; if

U a a X U a ar r� �1 1, . . . , ( , . . . , ) is an e.c.-set of the rank r, then � ( )U

g r�

1.

The function � can be continued to the algebra of cylindrical sets in a standard way.

Proposition 3. The function � is nonnegative, defined on algebra of c.-sets, and is countable-additive, i.e., is a

measure.

It is well known from the theory of measure [29] that there is a uniquely defined minimum �-algebra~S that contains

the algebra of cylindrical sets on which the measure � can be continued uniquely. Sets from the �-algebra~S are called

measurable. The function f defined on the space X is called measurable if it takes real values and the set of points x of

space X satisfying the inequality f x( ) � � is measurable for any real �. Let now S be a supplement of the �-algebra~S on

the measure �. The triple ( , , )X S � is called a stationary random process with discrete time. The above definition of a

stationary random process may be generalized in several directions:

a) a measurable space ( , )Y , where is a �-algebra of subsets on Y is taken instead of the alphabet X ;

b) the space A of sequences y y y y� �( . . . , , , , . . . )1 0 1 infinite in both sides is taken instead of space X ; and

c) cylindrical sets that define the �-algebra~S on A are determined as sets of the form

A y y y y y A y Y yi i i i ir r� � � �{ | ( . . . , , , . . . , , . . . ) : , . . . ,

1 2 1 1 �Yr },

where i ir1 , . . . , are integers, Yi � . Let � be a measure provided that � ( )A �1 on~S , and S be supplement of

~S in

this measure. The triple ( , , )A S � is called random process with discrete time. The stationarity condition for the

process is the independence of the measure of a cylindrical set from shift of all sequences of the given c.-set on a

531

finite number of positions to the left or to the right. The function defined on a random space and taking the values in

some measurable space is called random function. Problems similar to problems of sequence equidistribution occur also

for random functions [31].

The probabilistic theory of combinations, an interesting area of research, is represented in works by I. N. Kovalenko

and his disciples [32–34]. They study the asymptotic behavior of the sequences associated, in particular, with random logical

and random nonlinear Boolean equations, with arithmetical problems for the number of solutions of linear equations over

finite fields and finite rings, with random oriented and nonoriented graphs.

POLYNOMIALS, AUTOMATA, AND SEQUENCES

In this section, we present some applications and development of the above results and other directions of the studies.

We dwell on the following problem. Generation of random and pseudorandom sequences are considered in [35]. H. Shapiro

[36] introduced sequences, which now are called Rudin–Shapiro sequences [37]. Results of an analysis of such sequences, in

particular, polynomials that define them [38–40], have found an interesting application in applied problems of harmonic

analysis and in the theory of coding [38, 40]. Their generalization to an infinite case has resulted in introduction and

consideration of Shapiro series. It is interesting that the value of the sequence constructed using a Shapiro series is a

transcendental number. N. M. Korobov [14] and A. G. Postnikov [19] applied some arithmetical sequences to constructing

numerical methods of analysis, similar to the Monte Carlo method but with a reliable estimate of an error. One of the

directions of numerical mathematics, where various round-off errors are analyzed and taken into account, is interval analysis.

Note that joint use of estimates of deviation of some arithmetical sequence with estimates of variation of the function makes

it possible to obtain guaranteed interval estimates of integrals [41].

Some interesting automatic, semigroup, and group methods of generation of sequences are presented in [42]. Note

also that infinite sequences with interesting behavior may be associated with rather simple classes of polynomials. For

example, some sequences coupled to hyperelliptic curves and Artin–Schreier coverings are analyzed in [43] both

theoretically and experimentally. Concerning these sequences, it is either proved that they are uniformly distributed with

some density, in particular, with the Cato–Tate density, or assumed based on computer experiments that such type of

distribution should take place.

ELEMENTS OF ERGODIC THEORY

Let now ( , , )M � be a probability space on which the semigroup ( , , , , . . . )T kk � 0 1 2 of transforms of space M in

itself is defined, such that for any setU from the complete prototype U for the mapping T k belongs to , and � �T U U� �1

(invariance of a measure). In the ergodic theory, probability space together with the above-mentioned group of transforms

defined on it is called a dynamic system. The subset X M� is invariant if T X X1 � .

Further, we denote the dynamic system by the triple ( , , )M T� , unless otherwise specified. If the set M from

( , , )M T� cannot be presented as a sum of non-intersecting invariant sets of a positive measure, then the dynamic system

( , , )M T� is said to be irresolvable or ergodic. The Birkhoff ergodic theorem, proved also by Khinchin and reinforced by

Riesz [1], is widely applied in the theory of dynamic systems. Let us now recall the formulation of this theorem.

THEOREM C (Birkhoff–Khinchin–Riesz). Let � be an invariant measure. Let there be a function f x( ) absolutely

summable in the measure � (i.e., the integral

M

f x d� | ( )| � exists). Then:

a) for almost all (on the measure �) points x M� , the limit exists

lim( ) ( ) . . . ( )

( )n

nf x f Tx f T

ng x

��

� � �� ,

M M

g x d f x d� ��( ) ( )� �;

b) if the dynamic system is irresolvable in the measure �, then for almost all (on the measure �) points x M�

lim( ) ( ) . . . ( )

( )n

n

M

f x f Tx f T

nf x d

��

� � �� � �.

532

Here, we will restrict ourselves to dynamic systems whose space of states M is a space of sign sequence [1] (or, in

other words, M is a space of extra- (or super-) words) and some dynamic systems on Riemannian manifolds.

Dynamic Systems for Markov Chains [1]. Let p0 and p1 be two non-negative numbers and the matrix

p p

p ppij

00 01

10 11

0�

�� �

�� , ,

be formed by transition probabilities with the conditions p p00 01 1� � and p p10 11 1� � , and the stationarity

conditions p01� �p p p p0 01 0 10 , p p p p p10 1 01 1 10� � being fulfilled.

Let M be space of superwords in the {alpha}bet { }0 1, . On e.c. subsets U a ar( , . . . , )1 of the set M, the data presented

determine the measure � by the following rules:

1) � ( )M �1, � ( )O � 0, where O is an empty set; and

2) �U a a p p pr a a a a ar r( , . . . , ) , . . . ,1 1 1 2 1

��

.

Cylindrical sets are introduced as finite unions of e.c. sets. A fact similar to Proposition 1 holds that cylindrical sets

derivate an algebra. The measure � is continued from e.c. sets to the minimum �-algebra that contains the class of cylindrical

sets. Transforms on space M are determined as right shifts of T k by k positions:

T a a akk k� � �1 2 , . . . ,

and the measure � is invariant. The ergodicity condition for the dynamic system thus constructed is supplied by the

statement proved in [1].

Proposition 4. If the elements of the matrixp p

p p

00 01

10 11

�

�� �

��are positive, then the space M cannot be decomposed into a

sum of two invariant non-intersecting sets of a positive measure.

The concept of intermixing [30] is more general than the concept of ergodicity. The conversion T, appearing in the

definition of the dynamic system ( , , )M T� , is said to be (strong) intermixing if

lim ( ) ( ) ( )n

nA T B A B��

�� �� � �

for any A B, that belong to the �-algebra . As is generally known [30], the property of intermixing of a transform

yields ergodicity of this transform. Let for a connected aperiodic invertible Markov chain with a finite number n of its

states P Pxy� ( ) be a matrix of transition probabilities, where Pxy is the probability of transition from a state x into a

state y. For such matrix P, its degrees P t specify t-step transition probabilities, and for each y there exists the limit

� ( ) limy Pn

xyt�

��independent of the initial state x. By analogy with the problems of existence and construction of

efficient optimization procedures (see, for example, [44]), there appears a problem on existence of an efficient

intermixing, i.e., polynomial with respect to the length of the input (fast intermixing). One of the methods of solving

this problem is presented in [45]. Works by P. S. Knopov (see [46] and the bibliography therein) are devoted to the

intermixing criteria that provide in the limit independence, and the Markov behavior of the probability process.

SOME CONCEPTS OF THE RIEMANNIAN GEOMETRY

AND THEORY OF RIEMANNIAN SURFACES

Let us recall the definition of a Riemannian manifold. Let M be a smooth manifold of dimension n over a real field,

TM and T M* , respectively, be tangential and cotangent vector bundles over M. For vector bundle V, by C V� ( ) we denote

the vector space of smooth cuts of V. Elements C TM� ( ) and C T M� ( )* are called vector fields and 1-forms on M,

respectively. Using the operation of tensor product � with respect to the bundles TM and T M* , tensor bundles can be

constructed, and using the operation of external multiplication � with respect to T M* , external products � k T M* . Smooth

cuts of the bundle � k T M* are called k-dimensional differential forms. Let S T M2 * be a symmetric quadrate T M* (with

respect to a tensor product). By a Riemannian metric on M we mean a smooth cut of the bundle S T M2 * . In local coordinates

533

on M, the metric g is specified by the symmetric positive definite differential form � �g dx dxij i j , where g gij ji� are

functions from the open cover of M in the field R. In the above notation, the pair ( , )M g is called Riemannian variety. Let M

be a compact Riemannian variety with the metric of the above form. Such Riemannian metric specifies on M a measure

whose differential (the differential form that defines a measure) has the form d g dx dx dxij n� � � � � �det ( ) . . . ,1 2 .

Example 2 (ergodicity of shifts on tores). This simple example is well-known [29, 30]. Let Tn S S S� 1 1 1. . .

be the product of n circles.

Let � �1 , . . . , n be a set of real numbers, and for a point x x xn� ( , . . . , )1 from Tn , the shift transformation T is

specified by the formula

Tx x xn n� � �({ , . . . , )1 1� �} { } ,

where { }a is the fractional part of number a. Then for ergodicity of the transform T, it is necessary and sufficient that

the numbers 1 1, , . . . ,� �n be rationally independent.

Many problems and methods of the ergodic theory are motivated by problems of mechanics and physics, in brief, by

physical problems. One more line of development of the ergodic theory with the above motivation, and that motivated by

mathematical studies, involves action of semigroups and groups of transforms G on complex manifolds, M M S� ( ), defined on

compact Riemannian surfaces S of kind g. In terms of algebraic geometry, the surfaces S are not special algebraic curves of the

kind g, and M S( ) is an algebraic manifold associated with S , for example, a space of moduli. If S E� is an elliptic curve, i.e.,

is a non-special compact complex manifold of kind 1, then E can be associated in different ways with both hypothetical and

explicitly specified dynamic systems, some of which being ergodic [47–49]. If the kind g �1, then ergodicity holds in some

cases. Some results are presented below. We use concepts and results of complex analysis described in [50–52]. Only brief

explanations of some of them are presented below. Some computing aspects appearing in this case are presented in [53].

In formulating the results concerning ergodicity, Teichmuller spaces are used [51, 52]. Let us present some

explanations. Let f z( ) be a differentiable function of a complex variable with complex values. The conformal invariant is

associated with the function f

K ff f

f f

z z

z z

[ ]| / |

| / |�

� � �

� � ��

�� �

��1

1.

Here, z x iy� � , z x iy� � , and f z u z i z( ) ( ) ( )� � � , where u z( ) and � ( )z are functions taking on real values

� � �!!

� "!!

�

���

�

��� � � �

!!

� "!

fdf

dz

f

xi

f

yf

df

d z

f

xiz z

1

2

1

2,

f

y!�

���

�

��� .

Teichmuller considered extreme mapping min [ ]z S

zK f�

.

For analysis and evaluation of the above quantities, the following function is used:

� ( )/ , ,

, .z

f f z

z

z z�� � �

�#$%

Im

Im

0

0 0

The Beltrami differential equation �z z� � �/ is associated with the function � ( )z , which has a single-valued solution

provided that ( )� � �i i, � � �( )i 1. To operate with the Teichmuller space Tg , it is desirable to know the basis of

this space. The following theorem follows from the well-known Teichmuller theorem [51, 52].

THEOREM D. The set T of extreme quasiconformal mappings is the basis of the Teichmuller space.

The space of moduli of algebraic curves of the kind g is the factor &: M Tg g� / & of the Teichmuller space Tg with

respect to operation of modular group.

Ergodicity of the operation of a group of classes of surface transforms on the Naramsikhan–Sashardi bundles of a

universal moduli space is proved in [54]. And the operation of a group of classes of surface transforms on space of the first

group of surface cohomologies with the coefficients in some Lie group G is considered in [55]. If the surface is a closed

oriented Riemannian surface of a kind more than unity, and G is a compact connected Lie group, then such operation appears

ergodic [54, 55] . Due to intensive studies on quantum computers and quantum evaluations, the results described can be

applied to cybernetics and computer science, in particular, to quantum cryptography.

As indicated above and emphasized in [56, 57], some problems of arithmetical simulation of random processes and

the ergodic theory require an analysis for extremum during their solution. Efficient algorithms for solution of wide classes of

optimization problems are presented in [44].

534

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