  The conjugacy problem for the pseudo-Anosov automorphisms of a compact
surface is studied. To each pseudo-Anosov automorphism f, we assign an
AF-algebra A(f) (an operator algebra). It is proved that the assignment is
functorial, i.e. every f', conjugate to f, maps to an AF-algebra A(f'), which
is stably isomorphic to A(f). The new invariants of the conjugacy of the
pseudo-Anosov automorphisms are obtained from the known invariants of the
stable isomorphisms of the AF-algebras. Namely, the main invariant is a triple
(L, [I], K), where L is an order in the ring of integers in a real algebraic
number field K and [I] an equivalence class of the ideals in L. The numerical
invariants include the determinant D and the signature S, which we compute for
the case of the Anosov automorphisms. A question concerning the p-adic
invariants of the pseudo-Anosov automorphism is formulated.
