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