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Symbolic extensions and partially hyperbolic diffeomorphisms
1. | Dep. Matemática PUC-Rio, Marquês de São Vicente 225 22453-900, Rio de Janeiro, Brazil |
2. | Department of Mathematics, Brigham Young University, Provo, UT 84602, United States |
References:
[1] |
F. Abdenur, C. Bonatti, S. Crovisier, L. J. Díaz and G. Wen, Periodic points and homoclinic classes,, Ergod. Th. Dynamic. Systems, 27 (2007), 1.
|
[2] |
M. Asaoka, Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions,, Proc. Amer. Math. Soc., 136 (2008), 677.
doi: 10.1090/S0002-9939-07-09115-0. |
[3] |
C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms,, Ann. of Math., 143 (1996), 357.
doi: 10.2307/2118647. |
[4] |
C. Bonatti and L. J. Díaz, Connexions hétéroclines et généricité d'une infinité de puits ou de sources,, Ann. Scient. Éc. Norm. Sup., 32 (1999), 135.
|
[5] |
C. Bonatti and S. Crovisier, Recurrence et généricité,, Invent. Math., 158 (2004), 33.
|
[6] |
C. Bonatti and L. J. Díaz, On maximal transitive sets of generic diffeomorphisms,, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 171.
|
[7] |
C. Bonatti, L. J. Díaz and T. Fisher, Supergrowth of the number of periodic orbits for non-hyperbolic homoclinic classes,, Discrete and Cont. Dynamic. Systems, 20 (2008), 589.
|
[8] |
C. Bonatti, L. J. Díaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms; weak forms of hyperbolicity or infinitely many sinks or sources,, Annals of Math., 158 (2003), 355.
doi: 10.4007/annals.2003.158.355. |
[9] |
C. Bonatti, L. J. Díaz, and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,", volume \textbf{102} of Encyclopaedia of Mathematical Sciences, 102 (2005).
|
[10] |
C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting,, Israel J. Math., 115 (2000), 157.
doi: 10.1007/BF02810585. |
[11] |
Ch. Bonatti, L. J. Díaz, E.R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles,, Astérisque, 286 (2003), 187.
|
[12] |
M. Boyle and T. Downarowicz, Symbolic extension entropy: $C^r$ examples, products and flows,, Discrete Contin. Dyn. Syst., 16 (2006), 329.
doi: 10.3934/dcds.2006.16.329. |
[13] |
M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy, and subshift covers,, Forum Math., 14 (2002), 713.
doi: 10.1515/form.2002.031. |
[14] |
M. Brin and Stuck G, "Introduction to Dynamical Systems,", Cambridge University Press, (2002).
doi: 10.1017/CBO9780511755316. |
[15] |
D. Burguet, $C^2$ surface diffeomorphisms have symbolic extensions,, preprint, (). Google Scholar |
[16] |
K. Burns, personal, communication., (). Google Scholar |
[17] |
J. Buzzi, Intrinsic ergodicity for smooth interval maps,, Isreal J. Math., 100 (1997), 125.
doi: 10.1007/BF02773637. |
[18] |
J. Buzzi, T. Fisher, M. Sambarino and C. Vásquez, Intrinsic ergodicity for certain partially hyperbolic derived from Anosov systems,, preprint., (). Google Scholar |
[19] |
A. Candel and L. Conlon, "Foliations I,", volume \textbf{23} of Graduate Studies in Mathematics, 23 (2000).
|
[20] |
C. Carbalo, C. Morales and M. J. Pacifico, Homoclinic classes for generic $C^1$ vector fields,, Ergod. Th. Dynamic. Systems, 23 (2003), 403.
|
[21] |
W. Cowieson and L.-S. Young, SRB measures as zero-noise limits,, Ergod. Th. Dynamic. Systems, 25 (2005), 1115.
|
[22] |
L. J. Díaz, E. Pujals and R. Ures, Partial hyperbolicity and robust transitivity,, Acta Math., 183 (1999), 1.
doi: 10.1007/BF02392945. |
[23] |
T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions,, Inventiones Math., 176 (2009), 617.
doi: 10.1007/s00222-008-0172-4. |
[24] |
T. Downarowicz and S. Newhouse, Symbolic extensions in smooth dynamical systems,, Inventiones Math., 160 (2005), 453.
doi: 10.1007/s00222-004-0413-0. |
[25] |
N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations,, Discrete Contin. Dyn. Syst., 26 (2010), 1.
doi: 10.3934/dcds.2010.26.1. |
[26] |
S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\omega$-stability conjectures for flows,, Annals of Math., 145 (1997), 81.
doi: 10.2307/2951824. |
[27] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995).
|
[28] |
G. Keller, "Equilibrium States in Ergodic Theory,", London Mathematical Society Student Texts, (1998).
|
[29] |
R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383.
doi: 10.1016/0040-9383(78)90005-8. |
[30] |
M. Misiurewicz, Diffeomorphim without any measure of maximal entropy,, Bull. Acad. Pol. Sci., 21 (1973), 903.
|
[31] |
S. Newhouse, Hyperbolic limit sets,, Trans. Amer. Math. Soc., 167 (1972), 125.
doi: 10.1090/S0002-9947-1972-0295388-6. |
[32] |
S. Newhouse, Continuity properties of entropy,, Annals of Math., 129 (1989), 215.
doi: 10.2307/1971492. |
[33] |
M. J. Pacifico and J. Vieitez, Robust entropy-expansiveness implies generic domination,, preprint, ().
|
[34] |
M. J. Pacifico and J. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms,, Rev. Mat. Complut., 21 (2008), 293.
|
[35] |
E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Annals of Math., 151 (2000), 961.
doi: 10.2307/121127. |
[36] |
R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center,, Topology Appl., 157 (2010), 29.
doi: 10.1016/j.topol.2009.04.053. |
[37] |
M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27.
doi: 10.1090/S0002-9904-1974-13344-6. |
[38] |
M. Shub, "Global Stability of Dynamical Systems,", Springer-Verlag, (1987).
|
[39] |
K. Sigmund, Generic properties of invariant measures for Axiom-A-diffeomorphisms,, Inventiones Math., 11 (1970), 99.
doi: 10.1007/BF01404606. |
[40] |
C. P. Simon, Instability in Diff$(T^3)$ and the nongenericity of rational zeta function,, Trans. Amer. Math. Soc., 174 (1972), 217.
|
[41] |
P. Walters, "An Introduction to Ergodic Theory,", volume \textbf{79} of Graduate Texts in Mathematics, 79 (1982).
|
show all references
References:
[1] |
F. Abdenur, C. Bonatti, S. Crovisier, L. J. Díaz and G. Wen, Periodic points and homoclinic classes,, Ergod. Th. Dynamic. Systems, 27 (2007), 1.
|
[2] |
M. Asaoka, Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions,, Proc. Amer. Math. Soc., 136 (2008), 677.
doi: 10.1090/S0002-9939-07-09115-0. |
[3] |
C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms,, Ann. of Math., 143 (1996), 357.
doi: 10.2307/2118647. |
[4] |
C. Bonatti and L. J. Díaz, Connexions hétéroclines et généricité d'une infinité de puits ou de sources,, Ann. Scient. Éc. Norm. Sup., 32 (1999), 135.
|
[5] |
C. Bonatti and S. Crovisier, Recurrence et généricité,, Invent. Math., 158 (2004), 33.
|
[6] |
C. Bonatti and L. J. Díaz, On maximal transitive sets of generic diffeomorphisms,, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 171.
|
[7] |
C. Bonatti, L. J. Díaz and T. Fisher, Supergrowth of the number of periodic orbits for non-hyperbolic homoclinic classes,, Discrete and Cont. Dynamic. Systems, 20 (2008), 589.
|
[8] |
C. Bonatti, L. J. Díaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms; weak forms of hyperbolicity or infinitely many sinks or sources,, Annals of Math., 158 (2003), 355.
doi: 10.4007/annals.2003.158.355. |
[9] |
C. Bonatti, L. J. Díaz, and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,", volume \textbf{102} of Encyclopaedia of Mathematical Sciences, 102 (2005).
|
[10] |
C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting,, Israel J. Math., 115 (2000), 157.
doi: 10.1007/BF02810585. |
[11] |
Ch. Bonatti, L. J. Díaz, E.R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles,, Astérisque, 286 (2003), 187.
|
[12] |
M. Boyle and T. Downarowicz, Symbolic extension entropy: $C^r$ examples, products and flows,, Discrete Contin. Dyn. Syst., 16 (2006), 329.
doi: 10.3934/dcds.2006.16.329. |
[13] |
M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy, and subshift covers,, Forum Math., 14 (2002), 713.
doi: 10.1515/form.2002.031. |
[14] |
M. Brin and Stuck G, "Introduction to Dynamical Systems,", Cambridge University Press, (2002).
doi: 10.1017/CBO9780511755316. |
[15] |
D. Burguet, $C^2$ surface diffeomorphisms have symbolic extensions,, preprint, (). Google Scholar |
[16] |
K. Burns, personal, communication., (). Google Scholar |
[17] |
J. Buzzi, Intrinsic ergodicity for smooth interval maps,, Isreal J. Math., 100 (1997), 125.
doi: 10.1007/BF02773637. |
[18] |
J. Buzzi, T. Fisher, M. Sambarino and C. Vásquez, Intrinsic ergodicity for certain partially hyperbolic derived from Anosov systems,, preprint., (). Google Scholar |
[19] |
A. Candel and L. Conlon, "Foliations I,", volume \textbf{23} of Graduate Studies in Mathematics, 23 (2000).
|
[20] |
C. Carbalo, C. Morales and M. J. Pacifico, Homoclinic classes for generic $C^1$ vector fields,, Ergod. Th. Dynamic. Systems, 23 (2003), 403.
|
[21] |
W. Cowieson and L.-S. Young, SRB measures as zero-noise limits,, Ergod. Th. Dynamic. Systems, 25 (2005), 1115.
|
[22] |
L. J. Díaz, E. Pujals and R. Ures, Partial hyperbolicity and robust transitivity,, Acta Math., 183 (1999), 1.
doi: 10.1007/BF02392945. |
[23] |
T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions,, Inventiones Math., 176 (2009), 617.
doi: 10.1007/s00222-008-0172-4. |
[24] |
T. Downarowicz and S. Newhouse, Symbolic extensions in smooth dynamical systems,, Inventiones Math., 160 (2005), 453.
doi: 10.1007/s00222-004-0413-0. |
[25] |
N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations,, Discrete Contin. Dyn. Syst., 26 (2010), 1.
doi: 10.3934/dcds.2010.26.1. |
[26] |
S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\omega$-stability conjectures for flows,, Annals of Math., 145 (1997), 81.
doi: 10.2307/2951824. |
[27] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995).
|
[28] |
G. Keller, "Equilibrium States in Ergodic Theory,", London Mathematical Society Student Texts, (1998).
|
[29] |
R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383.
doi: 10.1016/0040-9383(78)90005-8. |
[30] |
M. Misiurewicz, Diffeomorphim without any measure of maximal entropy,, Bull. Acad. Pol. Sci., 21 (1973), 903.
|
[31] |
S. Newhouse, Hyperbolic limit sets,, Trans. Amer. Math. Soc., 167 (1972), 125.
doi: 10.1090/S0002-9947-1972-0295388-6. |
[32] |
S. Newhouse, Continuity properties of entropy,, Annals of Math., 129 (1989), 215.
doi: 10.2307/1971492. |
[33] |
M. J. Pacifico and J. Vieitez, Robust entropy-expansiveness implies generic domination,, preprint, ().
|
[34] |
M. J. Pacifico and J. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms,, Rev. Mat. Complut., 21 (2008), 293.
|
[35] |
E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Annals of Math., 151 (2000), 961.
doi: 10.2307/121127. |
[36] |
R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center,, Topology Appl., 157 (2010), 29.
doi: 10.1016/j.topol.2009.04.053. |
[37] |
M. Shub, Dynamical systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27.
doi: 10.1090/S0002-9904-1974-13344-6. |
[38] |
M. Shub, "Global Stability of Dynamical Systems,", Springer-Verlag, (1987).
|
[39] |
K. Sigmund, Generic properties of invariant measures for Axiom-A-diffeomorphisms,, Inventiones Math., 11 (1970), 99.
doi: 10.1007/BF01404606. |
[40] |
C. P. Simon, Instability in Diff$(T^3)$ and the nongenericity of rational zeta function,, Trans. Amer. Math. Soc., 174 (1972), 217.
|
[41] |
P. Walters, "An Introduction to Ergodic Theory,", volume \textbf{79} of Graduate Texts in Mathematics, 79 (1982).
|
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