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We present here a construction of horseshoes for any $\mathcal{C}^{1+\alpha}$mapping $f$ preserving an ergodic hyperbolic measure $\mu$ with $h_{\mu}(f)>0$and then deduce that the exponential growth rate of the number of periodicpoints for any $\mathcal{C}^{1+\alpha}$ mapping $f$ is greater than or equal to$h_{\mu}(f)$. We also prove that the exponential growth rate of the number ofhyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolicentropy means the entropy resulting from hyperbolic measures.

Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures