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

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

Yun Yang 1,

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.
Keywords: Mappings, hyperbolic measures, shadowing lemma, hyperbolic entropy., horseshoes.
Mathematics Subject Classification: 37B40, 37D25, 37C4.
Citation: Yun Yang. Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures. Discrete & Continuous Dynamical Systems, 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), 351-398. doi: 10.1007/s002220000057.  Google Scholar

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A. Avila, S. Crovisier and A. Wilkinson, Diffeomorphisms with positive metric entropy,, preprint, ().   Google Scholar

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M. Benedicks and M. Misiurewicz, Absolutely continuous invariant measures for maps with flat tops, Inst. Hautes Étides Sci. Publ. Math., 69 (1989), 203-213.  Google Scholar

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J. Buzzi, On the entropy-expanding maps,, preprint., ().   Google Scholar

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Y. M. Chung, Shadowing properties for non-invertible maps with hyperbolic measures, Tokyo J. Math., 22 (1999), 145-166. doi: 10.3836/tjm/1270041619.  Google Scholar

[6]

Y. M. Chung and M. Hirayama, Topological entropy and periodic orbits of saddle type for surface diffeomorphisms, Hiroshima Math. J., 33 (2003), 189-195.  Google Scholar

[7]

K. Gelfert, Expanding repellers for non-uniformly expanding maps with singularities and criticalities, Bull. Braz. Math. Soc., 41 (2010), 237-257. doi: 10.1007/s00574-010-0012-1.  Google Scholar

[8]

K. Gelfert and C. Wolf, On the distribution of periodic orbits, Discrete and Continuous Dynamical Systems, 26 (2010), 949-966. doi: 10.3934/dcds.2010.26.949.  Google Scholar

[9]

K. Gelfert and C. Wolf, Topological pressure via saddle orbits, Trans. Amer. Math. Soc., 360 (2008), 545-561. doi: 10.1090/S0002-9947-07-04407-8.  Google Scholar

[10]

A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.  Google Scholar

[11]

A. Katok and L. Mendoza, Supplement to Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[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), 203-219. doi: 10.1017/S0143385700001528.  Google Scholar

[13]

F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergod. Th. and Dynam. Sys, 1 (1981), 77-93.  Google Scholar

[14]

Z. Lian and L. S. Young, Lyapunov exponents, periodic orbits and Horseshoes for mappings of Hilbert spaces, Ann. Henri. Poincaré, 12 (2011), 1081-1108. doi: 10.1007/s00023-011-0100-9.  Google Scholar

[15]

C. Liang, G. Liao, W. Sun and X. Tian, Saturated set for nonuniformly hyperbolic systems,, preprint, ().   Google Scholar

[16]

G. Liao, W. Sun and X. Tian, Metric entropy and the number of periodic points, Nonlinearity, 23 (2010), 1547-1558. doi: 10.1088/0951-7715/23/7/002.  Google Scholar

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G. Liao, W. Sun and Y. Yang, Exponential rate of periodic points and metric entropy in nonuniformly hyperbolic systems,, preprint., ().   Google Scholar

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P. Liu, Stability of orbit spaces of mappings, Manuscripta Math., 93 (1997), 109-128. doi: 10.1007/BF02677460.  Google Scholar

[19]

P. Liu, Pesin's entropy formula for mappings, Nagoya Math. J., 150 (1998), 197-209.  Google Scholar

[20]

R. Mañe, Lyapunov exponents and invariant manifolds for compact transformations, in Geometric Dynamics, Lecture Notes in Math., 1007, Springer Verlag, 1983, 522-577. doi: 10.1007/BFb0061433.  Google Scholar

[21]

R. Mañe and C. Pugh, Stability of endomorphisms, in Dynamical Systems Warwick 1974, Lecture Notes in Math., 468, Springer, Berlin, 1975, 175-184.  Google Scholar

[22]

A. Manning, Topological entropy and the first homology group in Dynamical systems, in Dynamical Systems Warwick 1974, Lecture Notes in Mathematics, 468, Springer, Berlin, 1975, 185-190.  Google Scholar

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M. Misiurewicz and F. Przytcycki, Topological entropy and degree of smooth mappings, Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys., 25 (1977), 573-574.  Google Scholar

[24]

S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706, Springer Verlag, 1999.  Google Scholar

[25]

F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285.  Google Scholar

[26]

C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54. doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar

[27]

C. Pugh and M. Shub, Periodic points on the 2-Sphere, Discrete Contin. Dyn. Syst., 34 (2014), 1171-1182. doi: 10.3934/dcds.2014.34.1171.  Google Scholar

[28]

M. Qian, J. Xie and S. Zhu, Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics, 1978, Springer-Verlag, Berlin, 2009.  Google Scholar

[29]

D. Ruelle, An inequality for then entropy of differentiable maps, Bol. Soc. Bras. Math., 9 (1978), 83-87. doi: 10.1007/BF02584795.  Google Scholar

[30]

D. Ruelle, Characteristic exponents and invariant manifold in Hilbert space, Ann. of Math., 115 (1982), 243-290. doi: 10.2307/1971392.  Google Scholar

[31]

E. Sander, Homoclinic tangles for noninvertible maps, Nonlinear Analysis, 41 (2000), 259-276. doi: 10.1016/S0362-546X(98)00277-6.  Google Scholar

[32]

O. M. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, Journal of The American Mathematical Society, 26 (2013), 341-426. doi: 10.1090/S0894-0347-2012-00758-9.  Google Scholar

[33]

M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer.Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6.  Google Scholar

[34]

M. Shub and D. Sullivan, A remark on the lefschetz fixed point formula for differentiable maps, Topology, 13 (1974), 189-191. doi: 10.1016/0040-9383(74)90009-3.  Google Scholar

[35]

M. Shub, All, most, some differentiable dynamical systems, in International Congress of Mathematicians. Vol. III, European Math. Society, Zürich, 2006, 99-120.  Google Scholar

[36]

M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199. doi: 10.2307/2373276.  Google Scholar

show all references

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), 351-398. doi: 10.1007/s002220000057.  Google Scholar

[2]

A. Avila, S. Crovisier and A. Wilkinson, Diffeomorphisms with positive metric entropy,, preprint, ().   Google Scholar

[3]

M. Benedicks and M. Misiurewicz, Absolutely continuous invariant measures for maps with flat tops, Inst. Hautes Étides Sci. Publ. Math., 69 (1989), 203-213.  Google Scholar

[4]

J. Buzzi, On the entropy-expanding maps,, preprint., ().   Google Scholar

[5]

Y. M. Chung, Shadowing properties for non-invertible maps with hyperbolic measures, Tokyo J. Math., 22 (1999), 145-166. doi: 10.3836/tjm/1270041619.  Google Scholar

[6]

Y. M. Chung and M. Hirayama, Topological entropy and periodic orbits of saddle type for surface diffeomorphisms, Hiroshima Math. J., 33 (2003), 189-195.  Google Scholar

[7]

K. Gelfert, Expanding repellers for non-uniformly expanding maps with singularities and criticalities, Bull. Braz. Math. Soc., 41 (2010), 237-257. doi: 10.1007/s00574-010-0012-1.  Google Scholar

[8]

K. Gelfert and C. Wolf, On the distribution of periodic orbits, Discrete and Continuous Dynamical Systems, 26 (2010), 949-966. doi: 10.3934/dcds.2010.26.949.  Google Scholar

[9]

K. Gelfert and C. Wolf, Topological pressure via saddle orbits, Trans. Amer. Math. Soc., 360 (2008), 545-561. doi: 10.1090/S0002-9947-07-04407-8.  Google Scholar

[10]

A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.  Google Scholar

[11]

A. Katok and L. Mendoza, Supplement to Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[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), 203-219. doi: 10.1017/S0143385700001528.  Google Scholar

[13]

F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergod. Th. and Dynam. Sys, 1 (1981), 77-93.  Google Scholar

[14]

Z. Lian and L. S. Young, Lyapunov exponents, periodic orbits and Horseshoes for mappings of Hilbert spaces, Ann. Henri. Poincaré, 12 (2011), 1081-1108. doi: 10.1007/s00023-011-0100-9.  Google Scholar

[15]

C. Liang, G. Liao, W. Sun and X. Tian, Saturated set for nonuniformly hyperbolic systems,, preprint, ().   Google Scholar

[16]

G. Liao, W. Sun and X. Tian, Metric entropy and the number of periodic points, Nonlinearity, 23 (2010), 1547-1558. doi: 10.1088/0951-7715/23/7/002.  Google Scholar

[17]

G. Liao, W. Sun and Y. Yang, Exponential rate of periodic points and metric entropy in nonuniformly hyperbolic systems,, preprint., ().   Google Scholar

[18]

P. Liu, Stability of orbit spaces of mappings, Manuscripta Math., 93 (1997), 109-128. doi: 10.1007/BF02677460.  Google Scholar

[19]

P. Liu, Pesin's entropy formula for mappings, Nagoya Math. J., 150 (1998), 197-209.  Google Scholar

[20]

R. Mañe, Lyapunov exponents and invariant manifolds for compact transformations, in Geometric Dynamics, Lecture Notes in Math., 1007, Springer Verlag, 1983, 522-577. doi: 10.1007/BFb0061433.  Google Scholar

[21]

R. Mañe and C. Pugh, Stability of endomorphisms, in Dynamical Systems Warwick 1974, Lecture Notes in Math., 468, Springer, Berlin, 1975, 175-184.  Google Scholar

[22]

A. Manning, Topological entropy and the first homology group in Dynamical systems, in Dynamical Systems Warwick 1974, Lecture Notes in Mathematics, 468, Springer, Berlin, 1975, 185-190.  Google Scholar

[23]

M. Misiurewicz and F. Przytcycki, Topological entropy and degree of smooth mappings, Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys., 25 (1977), 573-574.  Google Scholar

[24]

S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706, Springer Verlag, 1999.  Google Scholar

[25]

F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285.  Google Scholar

[26]

C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54. doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar

[27]

C. Pugh and M. Shub, Periodic points on the 2-Sphere, Discrete Contin. Dyn. Syst., 34 (2014), 1171-1182. doi: 10.3934/dcds.2014.34.1171.  Google Scholar

[28]

M. Qian, J. Xie and S. Zhu, Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics, 1978, Springer-Verlag, Berlin, 2009.  Google Scholar

[29]

D. Ruelle, An inequality for then entropy of differentiable maps, Bol. Soc. Bras. Math., 9 (1978), 83-87. doi: 10.1007/BF02584795.  Google Scholar

[30]

D. Ruelle, Characteristic exponents and invariant manifold in Hilbert space, Ann. of Math., 115 (1982), 243-290. doi: 10.2307/1971392.  Google Scholar

[31]

E. Sander, Homoclinic tangles for noninvertible maps, Nonlinear Analysis, 41 (2000), 259-276. doi: 10.1016/S0362-546X(98)00277-6.  Google Scholar

[32]

O. M. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, Journal of The American Mathematical Society, 26 (2013), 341-426. doi: 10.1090/S0894-0347-2012-00758-9.  Google Scholar

[33]

M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer.Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6.  Google Scholar

[34]

M. Shub and D. Sullivan, A remark on the lefschetz fixed point formula for differentiable maps, Topology, 13 (1974), 189-191. doi: 10.1016/0040-9383(74)90009-3.  Google Scholar

[35]

M. Shub, All, most, some differentiable dynamical systems, in International Congress of Mathematicians. Vol. III, European Math. Society, Zürich, 2006, 99-120.  Google Scholar

[36]

M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199. doi: 10.2307/2373276.  Google Scholar

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