Time-symmetric Cycles

The main result of this note is that given any two time-symmetric cycles, one can find a time-symmetric extension of one by the other. This means that given a time-symmetric cycle, both time-symmetric doubles and square roots can be found. 1. Introduction. In [4], the idea of time-symmetric cycles was introduced. The basic idea underlying the definition of these cycles is that the dynamics associated to both the cycle and its inverse are isomorphic. This is a property that one would expect of physical data. Such data should not look more complicated if time is reversed. In [4], the classification of unimodal time-symmetric cycles was given. In this note, we drop the unimodal restriction and consider cycles in general. Section 2 gives the basic definitions. It is followed by some basic results concerning time-symmetric cycles; the numbers of time-symmetric cycles of length n are given as the number of diagonal points in a given cycle. It is then shown that given any two time-symmetric cycles, one can find a time-symmetric extension of one by the other. The note concludes by looking at the process of doubling and finding square roots.