Steklov Institute of Mathematics at St.Petersburg

PREPRINT 04/2010

ORBIT THEORY, LOCALLY FINITE PERMUTATIONS AND MORSE ARITHMETIC

This preprint was accepted April 14, 2010

ABSTRACT:
The goal of this paper is to analyze two measure preserving transformation
of combinatorial and number theoretical origin
from the point of view of ergodic orbit theory.
We study the Morse transformation (in its adic realization in the group $\textbf{Z}_2$ of
integer dyadic numbers -- \cite{V1,VA})
and prove that it has the same orbit partition as the dyadic odometer.
Then we give a precise description of time substitution of odometer 
which produces the Morse transformation.
 It is convenient to describe this time substitution in the form
 of random reordering of the group $\Bbb Z$, or in terms of 
random infinite permutations on the group $\Bbb Z$. 
We introduce the notion of {\it locally finite permutations} 
for the group  $\Bbb Z$ (and for all amenable groups.) 
Two automorphisms which have the same orbit partitions called 
{\it relational} if the time substitution of one to another is locally 
finite for almost all orbit. Our result is that Morse transformation 
and odometer are relational. 
The theory of random infinite permutations on the group $\Bbb Z$ 
(and on more general groups) is equally strong to the ergodic theory of action of the group. The main task in this area is the investigation of infinite permutations and the measures on it as well studying of linear orderings on $\Bbb Z$. 
 The class of locally finite permutations is a useful class for such analysis.
 

А. М. Вершик

ТРАЕКТОРНАЯ ТЕОРИЯ, ЛОКАЛЬНО-КОНЕЧНЫЕ ПОДСТАНОВКИ И АРИФМЕТИКА МОРСА

АННОТАЦИЯ
Вводится класс локально-конечных подстановок счетного множества и в
частности группы целых чисел,
который используется для нужд траекторной эргодической теории.
Доказываетя, что извесное пеобразование Морса
в ег адической реализации имеет то же разбиение на траектории, что и
одометр (=автоморфизм прибавления единицы
в аддитивной группе цеых 2-адических чисел), и что случайная замена
времени переводящая одометр в пробразование Морса - локально конечна с
вероятностью единица.