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October  2018, 38(10): 5011-5019. doi: 10.3934/dcds.2018219

On the hybrid control of metric entropy for dominated splittings

Xufeng Guo 1, , Gang Liao 2,*, , Wenxiang Sun 3, and  Dawei Yang 4,

1, 3. 

School of Mathematical Sciences, Peking University, Beijing 100871, China
2, 4. 

Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, China

* Author to whom any correspondence should be addressed

Received  October 2017 Revised  May 2018 Published  July 2018

Let $f$ be a $C^1$ diffeomorphism on a compact Riemannian manifold without boundary and $\mu$ an ergodic $f$-invariant measure whose Oseledets splitting admits domination. We give a hybrid estimate from above for the metric entropy of $\mu$ in terms of Lyapunov exponents and volume growth. Furthermore, for any $C^1$ diffeomorphism away from tangencies, its topological entropy is bounded by the volume growth.

Keywords: Metric entropy, Lyapunov exponent, volume growth.
Mathematics Subject Classification: 37D30, 37A35, 37D25.
Citation: Xufeng Guo, Gang Liao, Wenxiang Sun, Dawei Yang. On the hybrid control of metric entropy for dominated splittings. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5011-5019. doi: 10.3934/dcds.2018219
References:
[1]

F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60.  doi: 10.1007/s11856-011-0041-5.

[2]

A. Avila, S. Crovisier and A. Wilkinson, Diffeomorphisms with positive metric entropy, Publ. Math. Inst. Hautes Études Sci., 124 (2016), 319-347.  doi: 10.1007/s10240-016-0086-4.

[3]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Adv. Math., 226 (2011), 673-726.  doi: 10.4007/annals.2005.161.1423.

[4]

S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Adv. Math., 226 (2011), 673-726.  doi: 10.1016/j.aim.2010.07.013.

[5]

B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations, Electron. Res. Announc. Amer. Math. Soc., 3 (1997), 93-98.  doi: 10.1090/S1079-6762-97-00030-9.

[6]

M. Hirsch, C. Pugh and M. Shub, Invariant msnifolds, volume 583 of Lect. Notes in Math., Springer Verlag, 1977.

[7]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. 

[8]

O. Kozlovski, An integral formula for topological entropy of C∞ maps, Erg. Th. Dyn. Sys., 18 (1998), 405-424.  doi: 10.1017/S0143385798100391.

[9]

F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms. Part Ⅰ: Characterization of measures satisfying Pesin's entropy formula, Ann. of Math., 122 (1985), 509-539.  doi: 10.2307/1971328.

[10]

G. Liao, W. Sun and S. Wang, Upper semi-continuity of entropy map for nonnuiformly hyperbolic systems, Nonlinearity, 28 (2015), 2977-2992.  doi: 10.1088/0951-7715/28/8/2977.

[11]

G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, J. Eur. Math. Soc., 28 (2015), 2977-2992.  doi: 10.4171/JEMS/413.

[12]

S. Newhouse, Entropy and volume, Ergodic Theory Dynam. Systems, 8* (1988), 283–299. doi: 10.1017/S0143385700009469.

[13]

V. I. Oseledets, A multiplicative ergodic theorem, Trans. Moscow Math. Soc., 19 (1968), 179-210. 

[14]

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math.Surveys, 32 (1977), 55-112,287. 

[15]

F. Przytycki, An upper estimation for topological entropy of diffeomorphisms, Invent. Math., 59 (1980), 205-213.  doi: 10.1007/BF01453234.

[16]

D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat., 9 (1978), 83-88.  doi: 10.1007/BF02584795.

[17]

R. Saghin, Volume growth and entropy for C1 partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 34 (2014), 3789-3801.  doi: 10.3934/dcds.2014.34.3789.

[18]

W. Sun and X. Tian, Dominated splittings and Pesin's entropy formula, Discrete Contin. Dyn. Syst., 32 (2012), 1421-1434. 

[19]

P. Walters, An Introduction to Ergodic Theory, New York: Springer-Verlag, 1982.

[20]

J. Yang, C1 Dynamics far from Tangencies, PhD thesis, IMPA, Rio de Janeiro.

show all references

References:
[1]

F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60.  doi: 10.1007/s11856-011-0041-5.

[2]

A. Avila, S. Crovisier and A. Wilkinson, Diffeomorphisms with positive metric entropy, Publ. Math. Inst. Hautes Études Sci., 124 (2016), 319-347.  doi: 10.1007/s10240-016-0086-4.

[3]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Adv. Math., 226 (2011), 673-726.  doi: 10.4007/annals.2005.161.1423.

[4]

S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Adv. Math., 226 (2011), 673-726.  doi: 10.1016/j.aim.2010.07.013.

[5]

B. Hasselblatt and A. Wilkinson, Prevalence of non-Lipschitz Anosov foliations, Electron. Res. Announc. Amer. Math. Soc., 3 (1997), 93-98.  doi: 10.1090/S1079-6762-97-00030-9.

[6]

M. Hirsch, C. Pugh and M. Shub, Invariant msnifolds, volume 583 of Lect. Notes in Math., Springer Verlag, 1977.

[7]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. 

[8]

O. Kozlovski, An integral formula for topological entropy of C∞ maps, Erg. Th. Dyn. Sys., 18 (1998), 405-424.  doi: 10.1017/S0143385798100391.

[9]

F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms. Part Ⅰ: Characterization of measures satisfying Pesin's entropy formula, Ann. of Math., 122 (1985), 509-539.  doi: 10.2307/1971328.

[10]

G. Liao, W. Sun and S. Wang, Upper semi-continuity of entropy map for nonnuiformly hyperbolic systems, Nonlinearity, 28 (2015), 2977-2992.  doi: 10.1088/0951-7715/28/8/2977.

[11]

G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, J. Eur. Math. Soc., 28 (2015), 2977-2992.  doi: 10.4171/JEMS/413.

[12]

S. Newhouse, Entropy and volume, Ergodic Theory Dynam. Systems, 8* (1988), 283–299. doi: 10.1017/S0143385700009469.

[13]

V. I. Oseledets, A multiplicative ergodic theorem, Trans. Moscow Math. Soc., 19 (1968), 179-210. 

[14]

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math.Surveys, 32 (1977), 55-112,287. 

[15]

F. Przytycki, An upper estimation for topological entropy of diffeomorphisms, Invent. Math., 59 (1980), 205-213.  doi: 10.1007/BF01453234.

[16]

D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat., 9 (1978), 83-88.  doi: 10.1007/BF02584795.

[17]

R. Saghin, Volume growth and entropy for C1 partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 34 (2014), 3789-3801.  doi: 10.3934/dcds.2014.34.3789.

[18]

W. Sun and X. Tian, Dominated splittings and Pesin's entropy formula, Discrete Contin. Dyn. Syst., 32 (2012), 1421-1434. 

[19]

P. Walters, An Introduction to Ergodic Theory, New York: Springer-Verlag, 1982.

[20]

J. Yang, C1 Dynamics far from Tangencies, PhD thesis, IMPA, Rio de Janeiro.

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