Jump transformations and an embedding of O ∞ into O 2

A measurable map T on a measure space induces a representation ΠT of a Cuntz algebra ON when T satisfies a certain condition. For such two maps τ and σ and representations Πτ and Πσ associated with them, we show that Πτ is the restriction of Πσ when τ is a jump transformation of σ. Especially, the Gauss map τ1 and the Farey map σ1 induce representations Πτ1 of O∞ and that Πσ1 of O2, respectively, and Πτ1 = Πσ1 |O∞ with respect to a certain embedding of O∞ into O2. Mathematics Subject Classifications (2000). 11K50, 37C40, 46K10