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 [, p. 36] that an equivalent condition for ergodicity is thatfor 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. R. Matthews and A. M. Watts .
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 byNow 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.
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.
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 thatIn 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 .
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.