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

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 periodic points 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 of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.