We present here a construction of horseshoes for any C1+α mapping f preserving an ergodic hyperbolic measure μ with hμ(f)>0 and then deduce that the exponential growth rate of the number of periodic points for any C1+α mapping fis greater than or equal to hμ(f). We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.
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Yun Yang (2015) studied this question.
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