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Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics
Computational hyperbolicity
1. | Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków |
2. | Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Computer Science, Grota Roweckiego 6, 30-348 Kraków, Poland |
References:
[1] |
A. Al-Nayef, P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Bi-shadowing and delay equations, Dynam. Stability Systems, 10 (1996), 121-134. |
[2] |
A. A. Al-Nayef, P. E. Kloeden and A. V. Pokrovskii, Semi-hyperbolic mappings, condensing operators, and neutral delay equations, J. Differential Equations, 137 (1997), 320-339.
doi: 10.1006/jdeq.1997.3262. |
[3] |
Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, Experiment. Math., 16 (2007), 181-188. |
[4] | |
[5] |
R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458.
doi: 10.1016/0022-0396(74)90067-9. |
[6] |
M. Davis, R. MacKay and A. Sannami, Markov shifts in the Hénon family, Phys. D, 52 (1991), 171-178.
doi: 10.1016/0167-2789(91)90119-T. |
[7] |
S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka and P. Pilarczyk, Quantitative hyperbolicity estimates in one-dimensional dynamics, Nonlinearity, 21 (2008), 1967-1987.
doi: 10.1088/0951-7715/21/9/002. |
[8] |
R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping, Comm. Math. Phys., 67 (1979), 137-146.
doi: 10.1007/BF01221362. |
[9] |
P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings, J. Nonlinear Sci., 5 (1995), 419-431.
doi: 10.1007/BF01212908. |
[10] |
P. Diamond, P. E. Kloeden, V. S. Kozyakin and A. V. Pokrovskii, "Semi-Hyperbolicity and Bi-Shadowing,'', Manuscript., ().
|
[11] |
Z. Galias and P. Zgliczyński, Infinite-dimensional Krawczyk operator for finding periodic orbits of discrete dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4261-4272.
doi: 10.1142/S0218127407019937. |
[12] |
S. Hruska, A numerical method for constructing the hyperbolic structure of complex Hénon mappings, Found. Comput. Math., 6 (2006), 427-455.
doi: 10.1007/s10208-006-0141-2. |
[13] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Cambridge University Press, Cambridge, 1995. |
[14] |
M. Mazur, On some useful conditions on hyperbolicity, Trends in Math., 10 (2008), 57-64. |
[15] |
M. Mazur, J. Tabor and P. Kościelniak, Semi-hyperbolicity and hyperbolicity, Discrete Contin. Dynam. Syst., 20 (2008), 1029-1038.
doi: 10.3934/dcds.2008.20.1029. |
[16] |
M. Mazur, J. Tabor and K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case, Proceedings of the Conference "Topological Methods in Differential Equations and Dynamical Systems'' (Krakow-Przegorzaly, 1996), Univ. Iagell. Acta Math., 36 (1998), 121-126. |
[17] |
M. Mazur, J. Tabor, T. Kułaga and P. Kościelniak, Computational hyperbolicity group,, \url{http://www.im.uj.edu.pl/MarcinMazur/comphyp}., ().
|
[18] |
K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: A computer-assisted proof, Bull. Amer. Math. Soc. (N.S.), 32 (1995), 66-72. |
[19] |
S. Newhouse, Cone-fields, domination, and hyperbolicity, in "Modern Dynamical Systems and Applications,'' Cambridge Univ. Press, Cambridge, (2004), 419-432. |
[20] |
K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,'' Kluwer Academic Publishers, Dordrecht, 2000. |
[21] |
F. Riesz and B. Sz.-Nagy, "Functional Analysis,'' Frederick Ungar Publishing Co., New York, 1955. |
show all references
References:
[1] |
A. Al-Nayef, P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Bi-shadowing and delay equations, Dynam. Stability Systems, 10 (1996), 121-134. |
[2] |
A. A. Al-Nayef, P. E. Kloeden and A. V. Pokrovskii, Semi-hyperbolic mappings, condensing operators, and neutral delay equations, J. Differential Equations, 137 (1997), 320-339.
doi: 10.1006/jdeq.1997.3262. |
[3] |
Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, Experiment. Math., 16 (2007), 181-188. |
[4] | |
[5] |
R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458.
doi: 10.1016/0022-0396(74)90067-9. |
[6] |
M. Davis, R. MacKay and A. Sannami, Markov shifts in the Hénon family, Phys. D, 52 (1991), 171-178.
doi: 10.1016/0167-2789(91)90119-T. |
[7] |
S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka and P. Pilarczyk, Quantitative hyperbolicity estimates in one-dimensional dynamics, Nonlinearity, 21 (2008), 1967-1987.
doi: 10.1088/0951-7715/21/9/002. |
[8] |
R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping, Comm. Math. Phys., 67 (1979), 137-146.
doi: 10.1007/BF01221362. |
[9] |
P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings, J. Nonlinear Sci., 5 (1995), 419-431.
doi: 10.1007/BF01212908. |
[10] |
P. Diamond, P. E. Kloeden, V. S. Kozyakin and A. V. Pokrovskii, "Semi-Hyperbolicity and Bi-Shadowing,'', Manuscript., ().
|
[11] |
Z. Galias and P. Zgliczyński, Infinite-dimensional Krawczyk operator for finding periodic orbits of discrete dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4261-4272.
doi: 10.1142/S0218127407019937. |
[12] |
S. Hruska, A numerical method for constructing the hyperbolic structure of complex Hénon mappings, Found. Comput. Math., 6 (2006), 427-455.
doi: 10.1007/s10208-006-0141-2. |
[13] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Cambridge University Press, Cambridge, 1995. |
[14] |
M. Mazur, On some useful conditions on hyperbolicity, Trends in Math., 10 (2008), 57-64. |
[15] |
M. Mazur, J. Tabor and P. Kościelniak, Semi-hyperbolicity and hyperbolicity, Discrete Contin. Dynam. Syst., 20 (2008), 1029-1038.
doi: 10.3934/dcds.2008.20.1029. |
[16] |
M. Mazur, J. Tabor and K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case, Proceedings of the Conference "Topological Methods in Differential Equations and Dynamical Systems'' (Krakow-Przegorzaly, 1996), Univ. Iagell. Acta Math., 36 (1998), 121-126. |
[17] |
M. Mazur, J. Tabor, T. Kułaga and P. Kościelniak, Computational hyperbolicity group,, \url{http://www.im.uj.edu.pl/MarcinMazur/comphyp}., ().
|
[18] |
K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: A computer-assisted proof, Bull. Amer. Math. Soc. (N.S.), 32 (1995), 66-72. |
[19] |
S. Newhouse, Cone-fields, domination, and hyperbolicity, in "Modern Dynamical Systems and Applications,'' Cambridge Univ. Press, Cambridge, (2004), 419-432. |
[20] |
K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,'' Kluwer Academic Publishers, Dordrecht, 2000. |
[21] |
F. Riesz and B. Sz.-Nagy, "Functional Analysis,'' Frederick Ungar Publishing Co., New York, 1955. |
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