On the spectrum of anosov diffeomorphisms

Any Anosov diffeomorphismf induces a linear operator in the Banach space of continuous vector fields, which spectrum is contained in two annuli both being disjoint with the unit circle. The main theorem states that, if these annuli are sufficiently “narrow”, thenf has a fixed point. It is also proved that, iff acts on an infranilmanifold (e.g., on a torus), then the spectrum annuli for the algebraic automorphism corresponding tof are always more “narrow” than forf.