I. Sh. Jabbarov On Ergodic Hypothesis.

1. Introduction
   In the theory of stochastic processes The Ergodic Hypothesis of Boltzman (see [6, 7, 8, 14]) is well known. This hypothesis asserts that the average time of a finding of physical systems in some domain of phase space is equal to the relative measure of given domain (see [6], p. 701, [7], p. 522). However, the movement trajectory, in this case, depends on an initial position of a point, and the average on time undertaken irrespectively of it. The exact mathematical formulation of this hypothesis is expressed by a limit relation of a type
     ,) ( ) ( ) ( 1 lim 0          d f dt x T f T T t T (1) where tT some semigroup of mappings of phase space (see [6], [7]), x is, generally speaking, any point of the phase space. We will, at the expense of generality, to consider the problem in such statement. In [2, p. 457] this case is considered for continuous functions f in “a narrowed ergodic” case. Let’s notice that the Ergodic hypothesis, generally, is not true for the function (the example in [8] see, p. 536), accepting value 1 on a given trajectory, and 0, on other points of the phase space. Analogically, it is possible to construct a similar example in the infinite dimensional case. This is clear from the definition of product Lebesgue measure. In the works [6, 7, 8, 14] some have been proved ergodic theorems in which the relation (1) is proved for almost all points x of the phase space.
   In the present work we show that, if as a phase space to consider the cube      ]1,0 [ ]1,0 [ , it is possible to enter such a dynamical system for which the hypothesis mentioned above holds for a certain class of functions and any point x.

Let’s consider the dynamical system defined by a semigroup of mappings: ) (x x t  , where

2 }) ({ } { ) ( n n t t x t x x        , ,   x (the symbol {·} means a fractional part),   ( n ),   n 

is a sequence of positive numbers. Let f be a function defined in ,  ) , ( 0 2   L f where 0  designates the measures in the  introduced in the works [9, 11, 12, 13]. Function   ,… , ) ( 2 1    f f f   can be expanded into the Fourier series f ~ ) , ( 2 ) (  m i e m a  , where    , ,…) , ( 2 1 m m m  and
0  k m for all k m , with exception of finite number of them. Let for any natural N r  the multiple series
    r r Z m m m r m m m a ,…, , 2 1 2 1 ,…) 0, ,…, , (

converges (i.e. the corresponding subseries of given Fourier series converges absolutely). We will say that the function f belongs to the class  J if this condition satisfied for any natural r . Below we use the notions and designations defined in the works [9, 11, 12, 13].

Definition 1. Let N N  :  be any one to one mapping of the set of natural numbers. If for any m n > there is a natural number m such that n n = ) (  , then we call  a finite permutation. A subset   A is called to be finite-symmetrical if for any element A n  ) ( =   and any finite permutation  one has A n  ) (

) (    .

Let    and } | { ) (         . It is a countable set. We will designate ) (  the set of all limit points of the sequence ) (  . For every real t we write }) ({ } { n t t    . Let   ) , ( 2 ,… 1 ) , ( 2 1 ,…) 0, ,… ( ) ( ) (      m i Z m m r m i r e m m a e m a f r r      . The following theorem holds:

Theorem 1. There is a sequence of natural numbers  kr and a subset   0 of full measure such that the series     2 ) ( ) ( 1 k r r k k f f   converges everywhere in 0 .

Thus, the sequence   krf converges in 0  to some function ) ( f . The relation ) ( ) (   f f  is fair almoct everywhere in  and we will prove the relation (1) at first for the function ) ( ) (   f f  .

Theorem 2. For any function  J f the following relation holds                 ) ( ) ( ) ( ) ( 1 lim 0 0 0        d f d f dt t u f T T n T , for any point   u .

3

The increasing sequence of linearly independent frequencies n  plays here an essential role. In H. Bohr’s works [3, 4] averages of a kind (1) on the basis of Croneker’s theorem on the uniform distribution (mod1) of some curves in the multidimensional unite cube are studied. The theorem of
Croneker (see [18, p. 301]) states: Let N    ,…, , 2 1 be real numbers linearly independent over the field of rational numbers,  be a subdomain of N – dimensional unite cube with the volume  in Jordan meaning. Let further, ) (T I is a measure of a set of such ) ,0 ( T t  for which )1 (mod ) ,…, , ( 2 1      t t t N . Then . ) ( lim     T T I T 

Later the theory of uniform distribution of curves has been generalized and deeply studied by many other authors (see [15]). Generalization of this theory to the curves given by functions