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