# American Institute of Mathematical Sciences

October  2015, 35(10): 5133-5152. doi: 10.3934/dcds.2015.35.5133

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

 1 School of Mathematical Sciences, Peking University, Beijing, 100871, China

Received  December 2014 Revised  February 2015 Published  April 2015

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.
Citation: Yun Yang. Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5133-5152. doi: 10.3934/dcds.2015.35.5133
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