Jump transformations and an embedding of ${\cal O}_{\infty}$ into ${\cal O}_{2}$

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