A global volume lemma and applications

Let f ^t be a C ² Axiom A dynamical system on a compact manifold satisfying the transversality condition. We prove that if B _x (ε, t)=[y: dist (f ^s x,f ^s y)≤ε for all 0≤s≤t], then vol B _x (ε, t) has the order exp(∫ ₀ ^t φ (f ^s x)ds) in the continuous time case and exp (Σ _s ^t-1 φ (f ^s x)) in the discrete time case, where φ is a Holder continuous extension from basic hyperbolic sets of the negative of the differential expansion coefficient in the unstable direction. An application to the theory of large deviations is given.