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

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