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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

Ergodic theory is concerned with the extent to which iterates of a function mix subsets of a measure space. I will use the following basic concepts of ergodic theory specialized to the measure space with the measure . A measure-preserving function is ergodic if the only -measurable sets E for which are and the empty set, i.e., such a function does such a good job of mixing points in the space that it has no nontrivial -invariant sets. It can be shown [[39], p. 36] that an equivalent condition for ergodicity is that

for all and all integers . This condition in turn is equivalent to the assertion that for almost all the sequence of iterates

is uniformly distributed for all . A function is strongly mixing if

for all and all . Strongly mixing functions are ergodic.
The following result is a special case of a result of K. P. Matthews and A. M. Watts [51].

Theorem K

The map T is a measure-preserving transformation of which is strongly mixing. Consequently it is ergodic, and hence for almost all the sequence

is uniformly distributed for all .

Theorem K implies nothing about the behavior of T on the set of integers because it is a measure 0 subset of . In fact, the trajectory of any integer n can never have the property of the conclusion of Theorem K, for if the trajectory is eventually periodic with period k, it cannot be uniformly distributed , while if it is a divergent trajectory, it cannot even be equidistributed by (2.31). Consequently, this connection of the problem to ergodic theory does not seem to yield any deep insight into the problem itself.

There is, however, another connection of the problem to ergodic theory of that may conceivably yield more information on the problem. For each define the 0-1 variables by

Now define the function by , where


The value thus encodes the behavior of all the iterates of under T.
The following result has been observed by several people, including R. Terras and C. Pomerance, but has not been explicitly stated before.

Theorem L

The map is a continuous, one-one, onto, and measure-preserving map on the 2-adic integers .


The Conjecture can be reformulated in terms of the function as follows.

3x+1 CONJECTURE (Third form). Let denote the positive integers. Then . In fact .

For example , and .

The behavior of the function under iteration is itself of interest. Let denote the set of all rational numbers having odd denominators, so that . The set consists of exactly those 2-adic integers whose 2-adic expansion is finite or eventually periodic. The Finite Cycles Conjecture is equivalent to the assertion that there is a finite odd integer M such that

In fact one can take , where the product runs over all integers l for which there is a cycle of minimal length l. As a hypothesis for further work I advance the following conjecture.

For example, one may calculate that , , , . It can be shown that if n has a divergent trajectory, then the sequence cannot be eventually periodic. As a consequence the truth of the Periodicity Conjecture implies the truth of the Divergent Trajectories Conjecture.

Theorem B has a curious consequence concerning the fixed points of iterates of .

Theorem M

Suppose the kth iterate of has a fixed point which is not a fixed point of any for . Then k is a power of 2.


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