The study of the general behavior of the iterates of measure preserving
functions on a measure space is called ergodic theory.
The problem has some interesting connections to ergodic theory, because
the function T extends to a measure-preserving function on the 2-adic
integers defined with respect to the 2-adic measure.
To explain this, I need some basic facts about the 2-adic integers
,
cf. [14], [50].
The 2-adic integers consist of all series
where the are called the 2- adic digits of .
One can define congruences on
by
if the first k 2-adic digits of and agree.
Addition and multiplication on are given by
The 2- adic valuation on is given by and for
by , where is the
first nonzero 2-adic digit of .
The valuation induces a metric d on defined by
As a topological space is compact and complete with respect to the metric
d; a basis of open sets for this topology is given by the 2- adic discs of radius about :
Finally one may consistently define the 2- adic measure on so that
in particular .
The integers are a subset of ; for example
Now one can extend the definition of the function
given by (2.1) to by