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 map T is a measure-preserving transformation ofwhich is strongly mixing. Consequently it is ergodic, and hence for almost all
the sequence
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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
The mapis 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 iterateof
has a fixed point
which is not a fixed point of any
for
. Then k is a power of 2.