The behaviour of square functions from ergodic theory in $L^{\infty}$

In this paper, we analyze carefully the behaviour in $L^\infty(\mathbb R)$ of the square functions $S$ and $S_\mathcal I$'s, originating from ergodic theory. Firstly, we show that we can find some function $f\in L^\infty(\mathbb{R})$, such that $Sf$ equals infinity on a nonzero measure set. Secondly, we can find compact supported function $f\in L^\infty(\mathbb{R})$ and $\mathcal I$ such that $S_\mathcal{I} f$ does not belong to $BMO$ space. Finally, we show that $S$ is bounded from $L^{\infty}_c$ to $BMO$ space. As a consequence, we solve an open question posed by Jones, Kaufman, Rosenblatt and Wierdl in \cite{JKRW98}. That is, $S_\mathcal I$ are uniformly bounded in $L^p(\mathbb R)$ with respect to $\mathcal I$ for $2<p<\infty$.