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Global dynamics for symmetric planar maps
Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle
1. | LPMA Université Paris 6, 4 Place Jussieu, 75252 Paris Cedex 05, France |
2. | Department of Mathematics, Brigham Young University, Provo, UT 84602 |
References:
[1] |
M. Asaoka, Hyperbolic set exhibing $\mathcalC^1$-persistent homoclinic tangency for higher dimensions, Proc. Am. Math. Soc., 136 (2008), 677-686.
doi: 10.1090/S0002-9939-07-09115-0. |
[2] |
J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps, Ann. of Math. (2), 161 (2005), 1423-1485.
doi: 10.4007/annals.2005.161.1423. |
[3] |
R. Bowen, Entropy-expansive maps, Trans. Ame. Math. Soc., 164 (1972), 323-331. |
[4] |
M. Boyle and T. Downarowicz, The entropy theory of symbolic extension, Invent. Math., 156 (2004), 119-161 .
doi: 10.1007/s00222-003-0335-2. |
[5] |
M. Boyle and T. Downarowicz, Symbolic extension entropy : $\mathcalC^r$ examples, products and flows, Discrete Contin. Dyn. Syst., 16 (2006), 329-341.
doi: 10.3934/dcds.2006.16.329. |
[6] |
M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math., 14 (2002), 713-757.
doi: 10.1515/form.2002.031. |
[7] |
D. Burguet, $\mathcalC^2$ surface diffeomorphism have symbolic extensions, Invent. Math., 186 (2011), 191-236.
doi: 10.1007/s00222-011-0317-8. |
[8] |
D. Burguet, A direct proof of the variational principle for tail entropy and its extension to maps, Ergodic Theory Dynam. Systems, 29 (2009), 357-369.
doi: 10.1017/S0143385708080425. |
[9] |
D. Burguet, Symbolic extension for $\mathcalC^r$ non uniformly entropy expanding maps, Colloq. Math., 121 (2010), 129-151.
doi: 10.4064/cm121-1-12. |
[10] |
K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), 171 (2010), 451-489.
doi: 10.4007/annals.2010.171.451. |
[11] |
J. Buzzi, Intrinsic ergodicity for smooth interval maps, Israel J. Math., 100 (1997), 125-161.
doi: 10.1007/BF02773637. |
[12] |
W. Cowieson and L.-S. Young, SRB mesaures as zero-noise limits, Ergod. Th. Dynamic. Systems, 25 (2005), 1115-1138.
doi: 10.1017/S0143385704000604. |
[13] |
L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 29 (2011), 1419-1441.
doi: 10.3934/dcds.2011.29.1419. |
[14] |
L. J. Diaz, T. Fisher, M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., to appear, arXiv:1010.0721. |
[15] |
T. Downarowicz, "Entropy in Dynamical Systems, New Mathematical Monographs," 18, Cambridge University Press, Cambridge, 2011.
doi: 10.1017/CBO9780511976155. |
[16] |
T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116.
doi: 10.1007/BF02787825. |
[17] |
T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions: the antarctic theorem, Invent. Math., 176 (2009), 617-636.
doi: 10.1007/s00222-008-0172-4. |
[18] |
T. Downarowicz and S. Newhouse, Symbolic extensions in smooth dynamical systems, Invent. Math., 160 (2005), 453-499.
doi: 10.1007/s00222-004-0413-0. |
[19] |
M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes In Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. |
[20] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and Its Applications, 54, Cambridge University Press, Cambridge, 1995. |
[21] |
M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200. |
[22] |
S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.
doi: 10.2307/1971492. |
[23] |
V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 197-231. |
[24] |
M. Pacifico and J. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317. |
[25] |
Y. Pesin and L. Barreira, "Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Mathematics and Its Applications, 115, Cambridge University Press, Cambridge, 2007. |
[26] |
D. Ruelle, An inequality of the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.
doi: 10.1007/BF02584795. |
[27] |
M. Shub, "Global Stability of Dynamical Systems," With the collaboration of A. Fathi and R. Langevin. Transl. by J. Cristy, Springer-Verlag, New York, 1987. |
[28] |
P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
[29] |
Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.
doi: 10.1007/BF02766215. |
[30] |
Y. Yomdin, $\mathcalC^k$-resolution of semialgebraic mappings. Addendum to : "Volume growth and entropy", Israel J. Math., 57 (1987), 301-317.
doi: 10.1007/BF02766216. |
show all references
References:
[1] |
M. Asaoka, Hyperbolic set exhibing $\mathcalC^1$-persistent homoclinic tangency for higher dimensions, Proc. Am. Math. Soc., 136 (2008), 677-686.
doi: 10.1090/S0002-9939-07-09115-0. |
[2] |
J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps, Ann. of Math. (2), 161 (2005), 1423-1485.
doi: 10.4007/annals.2005.161.1423. |
[3] |
R. Bowen, Entropy-expansive maps, Trans. Ame. Math. Soc., 164 (1972), 323-331. |
[4] |
M. Boyle and T. Downarowicz, The entropy theory of symbolic extension, Invent. Math., 156 (2004), 119-161 .
doi: 10.1007/s00222-003-0335-2. |
[5] |
M. Boyle and T. Downarowicz, Symbolic extension entropy : $\mathcalC^r$ examples, products and flows, Discrete Contin. Dyn. Syst., 16 (2006), 329-341.
doi: 10.3934/dcds.2006.16.329. |
[6] |
M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math., 14 (2002), 713-757.
doi: 10.1515/form.2002.031. |
[7] |
D. Burguet, $\mathcalC^2$ surface diffeomorphism have symbolic extensions, Invent. Math., 186 (2011), 191-236.
doi: 10.1007/s00222-011-0317-8. |
[8] |
D. Burguet, A direct proof of the variational principle for tail entropy and its extension to maps, Ergodic Theory Dynam. Systems, 29 (2009), 357-369.
doi: 10.1017/S0143385708080425. |
[9] |
D. Burguet, Symbolic extension for $\mathcalC^r$ non uniformly entropy expanding maps, Colloq. Math., 121 (2010), 129-151.
doi: 10.4064/cm121-1-12. |
[10] |
K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), 171 (2010), 451-489.
doi: 10.4007/annals.2010.171.451. |
[11] |
J. Buzzi, Intrinsic ergodicity for smooth interval maps, Israel J. Math., 100 (1997), 125-161.
doi: 10.1007/BF02773637. |
[12] |
W. Cowieson and L.-S. Young, SRB mesaures as zero-noise limits, Ergod. Th. Dynamic. Systems, 25 (2005), 1115-1138.
doi: 10.1017/S0143385704000604. |
[13] |
L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 29 (2011), 1419-1441.
doi: 10.3934/dcds.2011.29.1419. |
[14] |
L. J. Diaz, T. Fisher, M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., to appear, arXiv:1010.0721. |
[15] |
T. Downarowicz, "Entropy in Dynamical Systems, New Mathematical Monographs," 18, Cambridge University Press, Cambridge, 2011.
doi: 10.1017/CBO9780511976155. |
[16] |
T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116.
doi: 10.1007/BF02787825. |
[17] |
T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions: the antarctic theorem, Invent. Math., 176 (2009), 617-636.
doi: 10.1007/s00222-008-0172-4. |
[18] |
T. Downarowicz and S. Newhouse, Symbolic extensions in smooth dynamical systems, Invent. Math., 160 (2005), 453-499.
doi: 10.1007/s00222-004-0413-0. |
[19] |
M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes In Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. |
[20] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and Its Applications, 54, Cambridge University Press, Cambridge, 1995. |
[21] |
M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200. |
[22] |
S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.
doi: 10.2307/1971492. |
[23] |
V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 197-231. |
[24] |
M. Pacifico and J. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317. |
[25] |
Y. Pesin and L. Barreira, "Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Mathematics and Its Applications, 115, Cambridge University Press, Cambridge, 2007. |
[26] |
D. Ruelle, An inequality of the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.
doi: 10.1007/BF02584795. |
[27] |
M. Shub, "Global Stability of Dynamical Systems," With the collaboration of A. Fathi and R. Langevin. Transl. by J. Cristy, Springer-Verlag, New York, 1987. |
[28] |
P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
[29] |
Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.
doi: 10.1007/BF02766215. |
[30] |
Y. Yomdin, $\mathcalC^k$-resolution of semialgebraic mappings. Addendum to : "Volume growth and entropy", Israel J. Math., 57 (1987), 301-317.
doi: 10.1007/BF02766216. |
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