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Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system
Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures
1. | School of Mathematical Sciences, Peking University, Beijing, 100871, China |
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, (). 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.
|
[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.
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, (). Google Scholar |
[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., (). Google Scholar |
[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. |
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.
doi: 10.1007/s002220000057. |
[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.
|
[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.
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, (). Google Scholar |
[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., (). Google Scholar |
[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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