On the hybrid control of metric entropy for dominated splittings

Previous Article
Selection of calibrated subaction when temperature goes to zero in the discounted problem
DCDS Home
This Issue
Next Article
Moduli of 3-dimensional diffeomorphisms with saddle-foci

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 & Continuous Dynamical Systems - A, 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Adv. Math., 226 (2011), 673-726. doi: 10.1016/j.aim.2010.07.013. Google Scholar
[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. Google Scholar
[6]	M. Hirsch, C. Pugh and M. Shub, Invariant msnifolds, volume 583 of Lect. Notes in Math., Springer Verlag, 1977. Google Scholar
[7]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. Google Scholar
[8]	O. Kozlovski, An integral formula for topological entropy of C^∞ maps, Erg. Th. Dyn. Sys., 18 (1998), 405-424. doi: 10.1017/S0143385798100391. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	S. Newhouse, Entropy and volume, Ergodic Theory Dynam. Systems, 8^* (1988), 283–299. doi: 10.1017/S0143385700009469. Google Scholar
[13]	V. I. Oseledets, A multiplicative ergodic theorem, Trans. Moscow Math. Soc., 19 (1968), 179-210. Google Scholar
[14]	Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math.Surveys, 32 (1977), 55-112,287. Google Scholar
[15]	F. Przytycki, An upper estimation for topological entropy of diffeomorphisms, Invent. Math., 59 (1980), 205-213. doi: 10.1007/BF01453234. Google Scholar
[16]	D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat., 9 (1978), 83-88. doi: 10.1007/BF02584795. Google Scholar
[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. Google Scholar
[18]	W. Sun and X. Tian, Dominated splittings and Pesin's entropy formula, Discrete Contin. Dyn. Syst., 32 (2012), 1421-1434. Google Scholar
[19]	P. Walters, An Introduction to Ergodic Theory, New York: Springer-Verlag, 1982. Google Scholar
[20]	J. Yang, C¹ Dynamics far from Tangencies, PhD thesis, IMPA, Rio de Janeiro.Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Adv. Math., 226 (2011), 673-726. doi: 10.1016/j.aim.2010.07.013. Google Scholar
[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. Google Scholar
[6]	M. Hirsch, C. Pugh and M. Shub, Invariant msnifolds, volume 583 of Lect. Notes in Math., Springer Verlag, 1977. Google Scholar
[7]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. Google Scholar
[8]	O. Kozlovski, An integral formula for topological entropy of C^∞ maps, Erg. Th. Dyn. Sys., 18 (1998), 405-424. doi: 10.1017/S0143385798100391. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	S. Newhouse, Entropy and volume, Ergodic Theory Dynam. Systems, 8^* (1988), 283–299. doi: 10.1017/S0143385700009469. Google Scholar
[13]	V. I. Oseledets, A multiplicative ergodic theorem, Trans. Moscow Math. Soc., 19 (1968), 179-210. Google Scholar
[14]	Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian Math.Surveys, 32 (1977), 55-112,287. Google Scholar
[15]	F. Przytycki, An upper estimation for topological entropy of diffeomorphisms, Invent. Math., 59 (1980), 205-213. doi: 10.1007/BF01453234. Google Scholar
[16]	D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat., 9 (1978), 83-88. doi: 10.1007/BF02584795. Google Scholar
[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. Google Scholar
[18]	W. Sun and X. Tian, Dominated splittings and Pesin's entropy formula, Discrete Contin. Dyn. Syst., 32 (2012), 1421-1434. Google Scholar
[19]	P. Walters, An Introduction to Ergodic Theory, New York: Springer-Verlag, 1982. Google Scholar
[20]	J. Yang, C¹ Dynamics far from Tangencies, PhD thesis, IMPA, Rio de Janeiro.Google Scholar

[1]	Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4767-4783. doi: 10.3934/dcds.2017205
[2]	Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
[3]	Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789
[4]	François Ledrappier, Seonhee Lim. Volume entropy of hyperbolic buildings. Journal of Modern Dynamics, 2010, 4 (1) : 139-165. doi: 10.3934/jmd.2010.4.139
[5]	Ilesanmi Adeboye, Harrison Bray, David Constantine. Entropy rigidity and Hilbert volume. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1731-1744. doi: 10.3934/dcds.2019075
[6]	Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048
[7]	Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197
[8]	Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003
[9]	Antonio Siconolfi, Gabriele Terrone. A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4409-4427. doi: 10.3934/dcds.2012.32.4409
[10]	Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249
[11]	Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215
[12]	Luis Barreira, Claudia Valls. Quadratic Lyapunov sequences and arbitrary growth rates. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 63-74. doi: 10.3934/dcds.2010.26.63
[13]	Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123
[14]	Frank Blume. Realizing subexponential entropy growth rates by cutting and stacking. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3435-3459. doi: 10.3934/dcdsb.2015.20.3435
[15]	César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577
[16]	Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921
[17]	Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131
[18]	Jairo Bochi, Michal Rams. The entropy of Lyapunov-optimizing measures of some matrix cocycles. Journal of Modern Dynamics, 2016, 10: 255-286. doi: 10.3934/jmd.2016.10.255
[19]	Alejo Barrio Blaya, Víctor Jiménez López. On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 433-466. doi: 10.3934/dcds.2012.32.433
[20]	Marcelo E. de Oliveira, Luiz M. G. Neto. Directional entropy based model for diffusivity-driven tumor growth. Mathematical Biosciences & Engineering, 2016, 13 (2) : 333-341. doi: 10.3934/mbe.2015005

American Institute of Mathematical Sciences