# 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
##### References:
 [1] J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding,, Invent. Math., 140 (2000), 351. doi: 10.1007/s002220000057. [2] A. Avila, S. Crovisier and A. Wilkinson, Diffeomorphisms with positive metric entropy,, preprint, (). [3] M. Benedicks and M. Misiurewicz, Absolutely continuous invariant measures for maps with flat tops,, Inst. Hautes Étides Sci. Publ. Math., 69 (1989), 203. [4] J. Buzzi, On the entropy-expanding maps,, preprint., (). [5] Y. M. Chung, Shadowing properties for non-invertible maps with hyperbolic measures,, Tokyo J. Math., 22 (1999), 145. doi: 10.3836/tjm/1270041619. [6] Y. M. Chung and M. Hirayama, Topological entropy and periodic orbits of saddle type for surface diffeomorphisms,, Hiroshima Math. J., 33 (2003), 189. [7] K. Gelfert, Expanding repellers for non-uniformly expanding maps with singularities and criticalities,, Bull. Braz. Math. Soc., 41 (2010), 237. doi: 10.1007/s00574-010-0012-1. [8] K. Gelfert and C. Wolf, On the distribution of periodic orbits,, Discrete and Continuous Dynamical Systems, 26 (2010), 949. doi: 10.3934/dcds.2010.26.949. [9] K. Gelfert and C. Wolf, Topological pressure via saddle orbits,, Trans. Amer. Math. Soc., 360 (2008), 545. doi: 10.1090/S0002-9947-07-04407-8. [10] A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms,, Publ. Math. IHES, 51 (1980), 137. [11] A. Katok and L. Mendoza, Supplement to Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, 54 (1995). doi: 10.1017/CBO9780511809187. [12] F. Ledrappier and J. M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula,, Ergod. Th. Dynam.Sys., 2 (1982), 203. doi: 10.1017/S0143385700001528. [13] F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval,, Ergod. Th. and Dynam. Sys, 1 (1981), 77. [14] Z. Lian and L. S. Young, Lyapunov exponents, periodic orbits and Horseshoes for mappings of Hilbert spaces,, Ann. Henri. Poincaré, 12 (2011), 1081. doi: 10.1007/s00023-011-0100-9. [15] C. Liang, G. Liao, W. Sun and X. Tian, Saturated set for nonuniformly hyperbolic systems,, preprint, (). [16] G. Liao, W. Sun and X. Tian, Metric entropy and the number of periodic points,, Nonlinearity, 23 (2010), 1547. doi: 10.1088/0951-7715/23/7/002. [17] G. Liao, W. Sun and Y. Yang, Exponential rate of periodic points and metric entropy in nonuniformly hyperbolic systems,, preprint., (). [18] P. Liu, Stability of orbit spaces of mappings,, Manuscripta Math., 93 (1997), 109. doi: 10.1007/BF02677460. [19] P. Liu, Pesin's entropy formula for mappings,, Nagoya Math. J., 150 (1998), 197. [20] R. Mañe, Lyapunov exponents and invariant manifolds for compact transformations,, in Geometric Dynamics, (1007), 522. doi: 10.1007/BFb0061433. [21] R. Mañe and C. Pugh, Stability of endomorphisms,, in Dynamical Systems Warwick 1974, (1974), 175. [22] A. Manning, Topological entropy and the first homology group in Dynamical systems,, in Dynamical Systems Warwick 1974, (1974), 185. [23] M. Misiurewicz and F. Przytcycki, Topological entropy and degree of smooth mappings,, Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys., 25 (1977), 573. [24] S. Y. Pilyugin, Shadowing in Dynamical Systems,, Lecture Notes in Math., (1706). [25] F. Przytycki, Anosov endomorphisms,, Studia Math., 58 (1976), 249. [26] C. Pugh and M. Shub, Ergodic attractors,, Trans. Amer. Math. Soc., 312 (1989), 1. doi: 10.1090/S0002-9947-1989-0983869-1. [27] C. Pugh and M. Shub, Periodic points on the 2-Sphere,, Discrete Contin. Dyn. Syst., 34 (2014), 1171. doi: 10.3934/dcds.2014.34.1171. [28] M. Qian, J. Xie and S. Zhu, Smooth Ergodic Theory for Endomorphisms,, Lecture Notes in Mathematics, (1978). [29] D. Ruelle, An inequality for then entropy of differentiable maps,, Bol. Soc. Bras. Math., 9 (1978), 83. doi: 10.1007/BF02584795. [30] D. Ruelle, Characteristic exponents and invariant manifold in Hilbert space,, Ann. of Math., 115 (1982), 243. doi: 10.2307/1971392. [31] E. Sander, Homoclinic tangles for noninvertible maps,, Nonlinear Analysis, 41 (2000), 259. doi: 10.1016/S0362-546X(98)00277-6. [32] O. M. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy,, Journal of The American Mathematical Society, 26 (2013), 341. doi: 10.1090/S0894-0347-2012-00758-9. [33] M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer.Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6. [34] M. Shub and D. Sullivan, A remark on the lefschetz fixed point formula for differentiable maps,, Topology, 13 (1974), 189. doi: 10.1016/0040-9383(74)90009-3. [35] M. Shub, All, most, some differentiable dynamical systems,, in International Congress of Mathematicians. Vol. III, (2006), 99. [36] M. Shub, Endomorphisms of compact differentiable manifolds,, Amer. J. Math., 91 (1969), 175. doi: 10.2307/2373276.

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##### References:
 [1] J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding,, Invent. Math., 140 (2000), 351. doi: 10.1007/s002220000057. [2] A. Avila, S. Crovisier and A. Wilkinson, Diffeomorphisms with positive metric entropy,, preprint, (). [3] M. Benedicks and M. Misiurewicz, Absolutely continuous invariant measures for maps with flat tops,, Inst. Hautes Étides Sci. Publ. Math., 69 (1989), 203. [4] J. Buzzi, On the entropy-expanding maps,, preprint., (). [5] Y. M. Chung, Shadowing properties for non-invertible maps with hyperbolic measures,, Tokyo J. Math., 22 (1999), 145. doi: 10.3836/tjm/1270041619. [6] Y. M. Chung and M. Hirayama, Topological entropy and periodic orbits of saddle type for surface diffeomorphisms,, Hiroshima Math. J., 33 (2003), 189. [7] K. Gelfert, Expanding repellers for non-uniformly expanding maps with singularities and criticalities,, Bull. Braz. Math. Soc., 41 (2010), 237. doi: 10.1007/s00574-010-0012-1. [8] K. Gelfert and C. Wolf, On the distribution of periodic orbits,, Discrete and Continuous Dynamical Systems, 26 (2010), 949. doi: 10.3934/dcds.2010.26.949. [9] K. Gelfert and C. Wolf, Topological pressure via saddle orbits,, Trans. Amer. Math. Soc., 360 (2008), 545. doi: 10.1090/S0002-9947-07-04407-8. [10] A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms,, Publ. Math. IHES, 51 (1980), 137. [11] A. Katok and L. Mendoza, Supplement to Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, 54 (1995). doi: 10.1017/CBO9780511809187. [12] F. Ledrappier and J. M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula,, Ergod. Th. Dynam.Sys., 2 (1982), 203. doi: 10.1017/S0143385700001528. [13] F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval,, Ergod. Th. and Dynam. Sys, 1 (1981), 77. [14] Z. Lian and L. S. Young, Lyapunov exponents, periodic orbits and Horseshoes for mappings of Hilbert spaces,, Ann. Henri. Poincaré, 12 (2011), 1081. doi: 10.1007/s00023-011-0100-9. [15] C. Liang, G. Liao, W. Sun and X. Tian, Saturated set for nonuniformly hyperbolic systems,, preprint, (). [16] G. Liao, W. Sun and X. Tian, Metric entropy and the number of periodic points,, Nonlinearity, 23 (2010), 1547. doi: 10.1088/0951-7715/23/7/002. [17] G. Liao, W. Sun and Y. Yang, Exponential rate of periodic points and metric entropy in nonuniformly hyperbolic systems,, preprint., (). [18] P. Liu, Stability of orbit spaces of mappings,, Manuscripta Math., 93 (1997), 109. doi: 10.1007/BF02677460. [19] P. Liu, Pesin's entropy formula for mappings,, Nagoya Math. J., 150 (1998), 197. [20] R. Mañe, Lyapunov exponents and invariant manifolds for compact transformations,, in Geometric Dynamics, (1007), 522. doi: 10.1007/BFb0061433. [21] R. Mañe and C. Pugh, Stability of endomorphisms,, in Dynamical Systems Warwick 1974, (1974), 175. [22] A. Manning, Topological entropy and the first homology group in Dynamical systems,, in Dynamical Systems Warwick 1974, (1974), 185. [23] M. Misiurewicz and F. Przytcycki, Topological entropy and degree of smooth mappings,, Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys., 25 (1977), 573. [24] S. Y. Pilyugin, Shadowing in Dynamical Systems,, Lecture Notes in Math., (1706). [25] F. Przytycki, Anosov endomorphisms,, Studia Math., 58 (1976), 249. [26] C. Pugh and M. Shub, Ergodic attractors,, Trans. Amer. Math. Soc., 312 (1989), 1. doi: 10.1090/S0002-9947-1989-0983869-1. [27] C. Pugh and M. Shub, Periodic points on the 2-Sphere,, Discrete Contin. Dyn. Syst., 34 (2014), 1171. doi: 10.3934/dcds.2014.34.1171. [28] M. Qian, J. Xie and S. Zhu, Smooth Ergodic Theory for Endomorphisms,, Lecture Notes in Mathematics, (1978). [29] D. Ruelle, An inequality for then entropy of differentiable maps,, Bol. Soc. Bras. Math., 9 (1978), 83. doi: 10.1007/BF02584795. [30] D. Ruelle, Characteristic exponents and invariant manifold in Hilbert space,, Ann. of Math., 115 (1982), 243. doi: 10.2307/1971392. [31] E. Sander, Homoclinic tangles for noninvertible maps,, Nonlinear Analysis, 41 (2000), 259. doi: 10.1016/S0362-546X(98)00277-6. [32] O. M. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy,, Journal of The American Mathematical Society, 26 (2013), 341. doi: 10.1090/S0894-0347-2012-00758-9. [33] M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer.Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6. [34] M. Shub and D. Sullivan, A remark on the lefschetz fixed point formula for differentiable maps,, Topology, 13 (1974), 189. doi: 10.1016/0040-9383(74)90009-3. [35] M. Shub, All, most, some differentiable dynamical systems,, in International Congress of Mathematicians. Vol. III, (2006), 99. [36] M. Shub, Endomorphisms of compact differentiable manifolds,, Amer. J. Math., 91 (1969), 175. doi: 10.2307/2373276.
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