  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.
