Metric entropy and the number of periodic points

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For an ergodic hyperbolic measure μ preserved by a C1+r(r > 0) diffeomorphism f, the exponential growth rate of the number of such periodic points that their atomic measures approximate μ and their Lyapunov exponents approximate the Lyapunov exponents of μ equals the metric entropy hμ(f) (see theorem 2.3). Moreover, this equality holds pointwise μ-a.e. (see theorem 2.4).