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On the hybrid control of metric entropy for dominated splittings
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 |
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.
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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