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
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