Cuntz–Krieger algebras representations from orbits of interval maps

Let f be an expansive Markov interval map with finite transition matrix Af. Then for every point, we yield an irreducible representation of the Cuntz–Krieger algebra OAf and show that two such representations are unitarily equivalent if and only if the points belong to the same generalized orbit. The restriction of each representation to the gauge part of OAf is decomposed into irreducible representations, according to the decomposition of the orbit.