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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 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

The mapTis a measure-preserving transformation of which is strongly mixing. Consequently it is ergodic, and hence for almost all the sequenceis 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 ofThe 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 .*

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

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 .

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

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