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

We present here a construction of horseshoes for any $\mathcal{C}^{1+α}$ mapping $f$ preserving an ergodic hyperbolic measure $μ$ with $h_μ(f)>0$ and then deduce that the exponential growth rate of the number of periodic points for any $\mathcal{C}^{1+α}$ mapping $f$ is greater than or equal to $h_μ(f)$. 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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