Using Ulam's method to calculate entropy and other dynamical invariants

Using a special form of Ulam's method, we estimate the measure-theoretic entropy of a triple (M, T. mu), where M is a smooth manifold, T is a C1+y uniformly hyperbolic map, and mu is the unique physical measure of T. With a few additional calculations, we also obtain numerical estimates of(i) the physical measure mu, (ii) the Lyapunov exponents of T with respect to Cc, (iii) the rate of decay of correlations for (T. mu) with respect to Cv test functions, and (iv) the rate of escape (for repellors). Four main situations are considered: T is everywhere expanding, T is everywhere hyperbolic (Anosov), T is hyperbolic on an attracting invariant set (axiom A attractor], and T is hyperbolic on a non-attracting invariant set (axiom A non-attractor/repellor).