Jump transformations and an embedding of ${\mathcal{O}}_{\mathrm{\infty }}$ into ${\mathcal{O}}_2$

A measurable map $T$ on a measure space induces a representation ${\mathrm{\Pi }}_T$ of a Cuntz algebra ${\mathcal{O}}_N$ when $T$ satisfies a certain condition. For such two maps $\tau$ and $\sigma$ and representations ${\mathrm{\Pi }}_{\tau }$ and ${\mathrm{\Pi }}_{\sigma }$ associated with them, we show that ${\mathrm{\Pi }}_{\tau }$ is the restriction of ${\mathrm{\Pi }}_{\sigma }$ when $\tau$ is a jump transformation of $\sigma$. Especially, the Gauss map ${\tau }_1$ and the Farey map ${\sigma }_1$ induce representations ${\mathrm{\Pi }}_{{\tau }_1}$ of ${\mathcal{O}}_{\mathrm{\infty }}$ and that ${\mathrm{\Pi }}_{{\sigma }_1}$ of ${\mathcal{O}}_2$, respectively, and $\Pi_{\tau_1}=\Pi_{\sigma_1}|{{\cal O}{\infty}}$withrespecttoacertainembeddingof${\cal O}{\infty}$into${\cal O}{2}$.