Uniform sets and super-stationary sets over general alphabets

Abstract

AbstractUniform sets and super-stationary sets over the binary alphabet have been extensively studied. In this paper, they are generalized to general alphabets. We generalize the fact that any uniform set contains a super-stationary set so that any uniform complexity is realized by a super-stationary set. This gives a formula to calculate the uniform complexity functions. We also give characterizations of the class of super-stationary sets in general settings in two somewhat different ways than in the binary case. Super-stationary sets are considered as phenomena which are independent of the time scale, but sensitive only to the direction of time, or dependent just on the order of events in time series. Hence, characterizations of super-stationary sets give insights into what is time, what looks like a history without a description of time duration, or what remains meaningful after we lose the quantitative sense of time.