John Franks and Bob Williams found an Anosov flow on a manifiold $M$ whose chain recorrent set is not all of $M$.
Under some pinching assumptions on the spectrum there are some answers. See Brin, M.; Manning, A.
Anosov diffeomorphisms with pinched spectrum. Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), pp. 48–53,
Lecture Notes in Math., 898, Springer, Berlin-New York, 1981.
See PORTEOUS, Hugh L. Anosov diffeomorphisms of flat manifolds. Topology, 1972, vol. 11, no 3, p. 307-315 for examples of manifolds admitting Anosov diffeomorphisms with first Betti number zero.