In some cases, Axiom A can be replaced by the condition that the diffeomorphism is a hyperbolic diffeomorphism on a hyperbolic set (Bowen 1975, Parry and Pollicott
1990).

Bowen, R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. New York: Springer-Verlag,
1975.Ott, E. Chaos
in Dynamical Systems. New York: Cambridge University Press, p. 143,
1993.Parry, W. and Pollicott, M. "Zeta Functions and the Periodic
Orbit Structure of Hyperbolic Dynamics." Astérisque No. 187-188,
1990.Smale, S. "Differentiable Dynamical Systems." Bull.
Amer. Math. Soc.73, 747-817, 1967.