Orbit representations and circle maps

We yield C*-algebras representations on the orbit spaces from the family of interval maps f (x) = beta x + alpha (mod 1) lifted to circle maps, in which case beta is an element of N. Each orbit will encode an unitary equivalence class of an irreducible representation of: a Cuntz algebra O-beta if alpha = 0 and beta > 1; an irrational rotation algebra A(beta) if alpha is not an element of Q and beta = 1; and a Cuntz-Krieger O-A alpha,O-beta whenever beta > 1 and the critical point is periodic, where A(alpha,beta) is the underlying Markov transition matrix of f.