Decomposition of Thompson group representations arising from Cuntz algebras

For every integer n >= 2 n 2 , we consider a family { pi w } w is an element of { 0 , 1 , .... , n - 1 } N of irreducible representations of the Cuntz algebra O n. All these representations (except one) are shown to be equivalent to those arising from the orbits of the interval map dynamical system ( I , f ) (I,f) with f ( x ) = n x ( mod 1 ) f(x)=nx 1) . We consider the embeddings V = V 2 subset of V n subset of O n V=V of the Thompson group in the Higman-Thompson group V n obtained by Birget and then from V n which was obtained independently by Birget and Nekrashevych. The restriction of pi w to V nis still irreducible; however, the restriction of pi w to is no longer irreducible, and we obtain the corresponding irreducible decomposition in terms of quasi-regular representations.