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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 .
[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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