A global volume lemma and applications

Letf ^t be aC ² Axiom A dynamical system on a compact manifold satisfying the transversality condition. We prove that ifB _x (ε,t)=y: dist (f ^s x,f ^s y)≤ε for all 0≤s≤t], then volB _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, whereφ 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. Partially supported by US-Israel BSF. Partially supported by a Darpa grant.