Connections of the problem to ergodic theory.

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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

if the first k 2-adic digits of

and

agree. Addition and multiplication on

are given by

The 2- adic valuation

is given by

and for

, 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

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

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

Now define the function

, 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 .

[Proof]

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.

PERIODICITY 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.

[Proof]

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