Full shifts and irregular sets

Abstract

By Birkhoff´s ergodic theorem, the set of points X for which the Birkhoff averages of a continuous function diverge has zero measure with respect to any finite invariant measure. Thus, at least from the point of view of ergodic theory, this set could not be smaller. Nevertheless, it can be large from other points of view. For example, for subshifts with the weak specification property, we showed recently, that X is residual whenever it is nonempty (it is a simple exercise to show that X is dense whenever it is nonempty). The main purpose of this note is to convey in the simplest possible manner the proof of our result in the particular case of the full shift on a finite number of symbols. This has the advantage of avoiding some accessory technicalities that are necessary in the general case. In fact, we consider also the more general case when the set of accumulation points of the Birkhoff averages of a continuous function is prescribed closed interval and we show that it is residual whenever it is nonempty.