Endomorphisms of B(H). II. Finitely Correlated States on On

We identify sets of conjugacy classes of ergodic endomorphisms of B(H) where H is a fixed separable Hilbert space. They correspond to certain equivalence classes of pure states on the Cuntz algebras Onwherenis the Powers index. These states, called finitely correlated states, and strongly asymptotically shift invariant states, are defined and characterized. The subsets of these states defining shifts will in general be identified in a later work, but here an interesting cross section for the conjugacy classes of shifts called diagonalizable shifts is introduced and studied.