Using Ulam’s method to calculate entropy and other dynamical invariants
نویسنده
چکیده
Using a special form of Ulam’s method, we estimate the measure-theoretic entropy of a triple (M, T , μ), where M is a smooth manifold, T is a C1+γ uniformly hyperbolic map, and μ is the unique physical measure of T . With a few additional calculations, we also obtain numerical estimates of (i) the physical measureμ, (ii) the Lyapunov exponents of T with respect toμ, (iii) the rate of decay of correlations for (T , μ)with respect to C 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). AMS classification scheme numbers: 28D20, 41A25, 58F11, 58F19
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