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\begin{document}

\title{On the structure of invariant ergodic measures related with a class
of ergodic discrete dynamical systems }
\author{A.M. Samoylenko\thanks{%
Mailling~address:~Institute Mathematics of NAS, Kyiv,~00601,~Ukraine}, Y.A.
Prykarpatsky\thanks{%
Mailling~address:~Institute Mathematics of NAS, Kyiv,~00601,~Ukraine} and A.
K. Prykarpatsky\thanks{%
Dept. of Applied Mathematics, the AGH University of Science and Technology,
Cracow,\ 30059, Poland, and Institute for Applied Problem of Mechanics and
Mathematics, National Academy of Sciences{}, Lviv, 79601 Ukraine} }
\date{\today}
\maketitle

\begin{abstract}
\noindent We study the problem of constructing an invariant ergodic measure
naturally related with an ergodic discrete system on a topological phase
space \ endowed with a structure of measurable space. Based on the classical
Hardy results on functions series convergence we found an invariant ergodic
measure represented in a smeared form of Lebesgue-Stiltjes type measures on
the circle.
\end{abstract}

\section{Introduction}

Suppose that a topological phase space \ $M$ \ is endowed with a structure
of measurable space, that is a \ $\sigma $- algebra \ $A$($M$) \ of subsets
in \ $M$, \ on which there is defined a finite measure $\mu $ : $%
A(M)\longrightarrow \mathbb{R}_{\dotplus },$ $\mu (M)=$1. \ As is \ well
known, a measurable mapping \ $\varphi $ : $M\longrightarrow $ $M$ of the
measurable space $(M,A(M))$ \ is called ergodic discrete dynamic system, if $%
\mu $ - almost everywhere ($\mu $ - a.e.) there exists the not depending on $%
x$ $\in M$ limit \ lim$_{n\longrightarrow \infty }\frac{\text{1}}{\text{2}}%
\sum_{k=0}^{n-1}f(\varphi ^{k}x):=f_{\varphi }(x)$ \ for \ any bounded
measurable function $f\in \mathcal{B}(M;\mathbb{R)},$ defining for all \ $%
x\in M$ a finite measure \ $\mu _{\varphi }:A$($M$)$\longrightarrow \mathbb{R%
}_{\dotplus }$ on $M$, such that \ $\int_{M}f_{\varphi }(x)d\mu
(x):=\int_{M}f(x)d\mu _{\varphi }(x)$. \ The measure \ $\mu _{\varphi }:A$($%
M $)$\longrightarrow \mathbb{R}_{\dotplus }$ \ defined \ above has the
following invariance property subject to the dynamical system $\ \varphi
:M\longrightarrow M:\mu _{\varphi }(\varphi ^{-1}A)=\mu _{\varphi }(A)$ for
any \ $A\in A(M)$. \ Moreover, if a \ $\sigma $- measurable set \ $A\in A(M)$
\ is invariant with respect to the mapping \ $\varphi :M\longrightarrow M$,
that \ is $\varphi ^{-1}(A)=A$, \ then evidently $\mu _{\varphi }(A)=\mu (A)$%
. Therefore the existence of the \ $\varphi $-invariant measure \ $\mu
_{\varphi }:A$($M$)$\longrightarrow \mathbb{R}_{\dotplus }$, \ coinciding
with the measure $\mu :A$($M$)$\longrightarrow \mathbb{R}_{\dotplus }$ on
the $\sigma $ - algebra \ $\mathcal{I}(M)\subset A(M)$ \ of invariant (with
respect to the dynamical system $\varphi :M\longrightarrow M)$ sets, is a
necessary condition of the convergence $\mu $ - a.e. at $x\in M$ \ of mean
values $f_{\varphi }(x)$ for any $f\in \mathcal{B}(M;\mathbb{R)}$. The
inverse is also true: due to the Birkhoff theorem, if the mapping $\varphi
:M\longrightarrow M$ conserves a finite measure \ $\mu _{\varphi
}:A(M)\longrightarrow \mathbb{R}_{\dotplus }$, mean values \ $f_{\varphi
}(x) $ \ are convergent \ $\mu _{\varphi }$ - a.e. on $M$, and the set of
convergency is invariant. \ Thus, if the reduction of the measure \ $\mu :A$(%
$M$)$\longrightarrow \mathbb{R}_{\dotplus }$ \ upon the invariant $\sigma $
- algebra $\ \mathcal{I}(M)\subset A(M)$ is absolutely continuous with
respect to that of the measure \ $\mu _{\varphi }:A(M)\longrightarrow 
\mathbb{R}_{\dotplus }$ , then the convergence holds $\mu $ - a.e. on $M$ .
\ Thereby the problem of retrieving in analytical form the invariant measure 
$\mu _{\varphi }:A(M)\longrightarrow \mathbb{R}_{\dotplus }$ \ corresponding
to an ergodic \ mapping \ $\varphi :M\longrightarrow M$ \ is of great
interest both from theoretical and practical points of view. Our article we
devote to developing a measure generating function approach to constructing
the invariant measure based on classical real analysis of functional series.

Assume we are given a discrete dynamical system $\varphi :M\longrightarrow M$
. \ Then one can define measure generating functions (m.g.f.) $\mu _{\varphi
}(\lambda ;A):=$lim$_{n\longrightarrow \infty }\sum_{k=0}^{n-1}\lambda
^{k}\mu (\varphi ^{-k}A)$ , where \ $A\in A(M)$ \ and $|\lambda |<1$.

\textbf{Lemma. \ }\textit{The m.g.f. satisfies the functional equation \ }$%
\mu _{\varphi }(\lambda ;A)=\lambda \mu _{\varphi }(\lambda ;\varphi
^{-1}A)+\mu (A)$ \textit{for any }\ $A\in A(M)$ \textit{and} \ $|\lambda |<1$%
.

Based on this Lemma the following statements are proved:

\textbf{Theorem 1. }\textit{Let the m.g.f. \ \ }$\mu _{\varphi }:$\textit{\ }%
$\mathbb{C}\times $\textit{\ }$A(M)\longrightarrow \mathbb{C}$ ,\textit{\
corresponding to a discrete dynamical system \ }$\varphi :M\longrightarrow M$%
, \textit{\ \ exist and satisfy the invariance condition. Then the limit
expression \ }lim$_{\lambda \upharpoonright 1\text{ }(\mathcal{Q}\lambda
=0)}(1-\lambda )\mu _{\varphi }(\lambda ;A)=\mu _{\varphi }(A)$\textit{\
holds \ for any \ }$A\in A(M).$

It is seen that the series $\mu _{\varphi }(\lambda ;A)$ generates \
analytical \ \ function as $\ |\lambda |<1$, having the following property:
for any \ $\lambda \in (-1,1)$ \ and $A\in A(M)$ \ \ $\mathcal{Q}\mu
_{\varphi }(\lambda ;A)=0$. Based now on classical analytical functions
theory results one can formulate the main theorem.

\textbf{Theorem 2. \ }\textit{Let a \ m.g.f. }$\mu _{\varphi }:$\textit{\ }$%
\mathbb{C}\times $\textit{\ }$A(M)\longrightarrow \mathbb{C}$ \ \textit{%
satisfy the above conditions. Then the following representation holds: \ }$%
\mu _{\varphi }(\lambda ;A)=\int_{0}^{2\pi }\frac{(1-\lambda ^{2})d\sigma
_{\varphi }(s;A)}{1-2\lambda \cos s+\lambda ^{2}}$ \textit{for \ any }$A\in
A(M),$\ where $\sigma _{\varphi }($o$;A):[0,2\pi ]\rightarrow \mathbb{R}%
_{\dotplus }$ \textit{\ is a \ function of bounded variation: \ }$0\leq
\sigma _{\varphi }(s;A)\leq \mu (A)$ \textit{for any }$s\in \lbrack 0,2\pi ]$
\textit{and }$A\in A(M).$

This theorem looks exceptionally interesting for applications since it
reduces the problem of detecting the invariant measure \ $\mu _{\varphi
}:A(M)\longrightarrow \mathbb{R}_{\dotplus }$ defined by (1.6) to
calculation of the following complex analytical limit : \textit{\ }$\mu
_{\varphi }(A)=lim_{\lambda \upharpoonright 1(\lambda =0)}\int_{0}^{2\pi }%
\frac{2(1-\lambda ^{2})d\sigma _{\varphi }(s;A)}{1-2\lambda \cos s+\lambda
^{2}}$ , \ where $A\in A(M)$ and \ $\sigma _{\varphi }:[0,2\pi ]\times
A(M)\rightarrow \mathbb{R}_{\dotplus }$ - some Stiltjes measure on $[0,2\pi
] $ , \ generated by a given \textit{a priori} dynamical \ system $\varphi
:M\rightarrow M$ \ and a measure $\mu :A(M)\rightarrow \mathbb{R}_{\dotplus
} $. For instance, in the case of the well known Gauss mapping \ $\varphi
:M\rightarrow M$ , where $M=(0,1)$ and for any $x\in (0,1)$ \ $\varphi
(x):=\{1/x\},$ \ ( here "\{o\}" \ means taking the fractional part of a
number \ $x\in (0,1)),$ one can show by means of simple but a little
cumbersome calculations, that its invariant measure on $\ M$ is given by the
expression $\mu _{\varphi }(A)=\frac{1}{\ln 2}\int_{A}\frac{dx}{1+x}$ , \
yielding obviously, the well known Gauss measure $\mu _{\varphi
}:A(M)\longrightarrow \mathbb{R}_{\dotplus }$ on $M=(0,1)$. As a result ,
the following limit \ lim$_{n\longrightarrow \infty
}\sum_{k=0}^{n-1}f(\varphi ^{n}x)\overset{\text{a.e. }}{=}\frac{1}{\ln 2}%
\int_{0}^{1}f(x)/(1+x)dx$ for arbitrary $f\in L_{1}(0,1)$ \ is true. The
analytical expression obtained above for the invariant measure $\mu
_{\varphi }:A(M)\longrightarrow \mathbb{R}_{\dotplus }$ , \ generated by a
discrete dynamical system $\varphi :M\longrightarrow M$, looks interesting
for applications. In particular, it is simply derived from (3.4) that the
Stiltjes measure $\sigma _{\varphi }($o$;A):[0,2\pi ]\longrightarrow \lbrack
0,\mu (A)],A\in A(M),$ generates for any $s\in \lbrack 0,2\pi ]$ \ a new
positive definite measure on $A\in A(M)$ as $\sigma _{\varphi }(s)(A)=\sigma
_{\varphi }(s;A),$ which can regarded as the measure \ $\mu
:A(M)\longrightarrow \mathbb{R}_{\dotplus }$ smeared along the unit circle $%
\ S^{1\text{ \ }}$in the $\func{complex}$ plane $\mathbb{C}$.

\ \ \ \ \textbf{References:}

\bigskip\ \ \ \ \ [1] \ Wheedon R., Zygmund A. Measure and integral: an
introdaction to real analysis. Marcel Decker,

\ \ \ \ \ \ \ \ \ \ Inc., NY, \ \ and Basel,1977.

\ \ \ \ [2] \ Sinai Ya.G. Ergodic theory. Nauka Publ., Moscow, 1984 (in
Russian)

\ \ \ \ [3] \ Hardy G., Convergent series., Cambridge Press, 1947.

\ \ \ \ [4] \ Polya G., Sego H. Problems and solutios. Springer, Ny, 1982.

\ \ \ \ [5] \ Privalov I.I. Boundary properties of analytical functions.
Gostekhizdat, Publ., Moscow,1950.

\ \ \ \ [6] \ Skorokhod A.V. Elements of the probability theory and czsual
processes. Vyshcha Shkola Publ., Kyiv,

\ \ \ \ \ \ \ \ 1975.

\ \ \ \ \

\ \ \ \

\end{document}