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October  2011, 29(4): 1419-1441. doi: 10.3934/dcds.2011.29.1419

Symbolic extensions and partially hyperbolic diffeomorphisms

Lorenzo J. Díaz 1, and  Todd Fisher 2,

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

Received  November 2009 Revised  August 2010 Published  December 2010

We show there are no symbolic extensions $C^1$-generically among diffeomorphisms containing nonhyperbolic robustly transitive sets with a center indecomposable bundle of dimension at least 2. Similarly, $C^1$-generically homoclinic classes with a center indecomposable bundle of dimension at least 2 that satisfy a technical assumption called index adaptation have no symbolic extensions.
Keywords: partially hyperbolic, homoclinic tangency., dominated splitting, Symbolic extensions, robust transitivity.
Mathematics Subject Classification: Primary: 37D30, 37C05, 37C20, 37B1.
Citation: Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419
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), 1-22.

[2]

M. Asaoka, Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions, Proc. Amer. Math. Soc., 136 (2008), 677-686. doi: 10.1090/S0002-9939-07-09115-0.

[3]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396. 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), 135-150.

[5]

C. Bonatti and S. Crovisier, Recurrence et généricité, Invent. Math., 158 (2004), 33-104.

[6]

C. Bonatti and L. J. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 171-197.

[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), 589-604.

[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), 355-418. 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 102 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2005.

[10]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193. 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), 187-222.

[12]

M. Boyle and T. Downarowicz, Symbolic extension entropy: $C^r$ examples, products and flows, Discrete Contin. Dyn. Syst., 16 (2006), 329-341. 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), 713-757. 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, (). 

[16]

K. Burns, personal, communication., (). 

[17]

J. Buzzi, Intrinsic ergodicity for smooth interval maps, Isreal J. Math., 100 (1997), 125-161. 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., (). 

[19]

A. Candel and L. Conlon, "Foliations I," volume 23 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2000.

[20]

C. Carbalo, C. Morales and M. J. Pacifico, Homoclinic classes for generic $C^1$ vector fields, Ergod. Th. Dynamic. Systems, 23 (2003), 403-415.

[21]

W. Cowieson and L.-S. Young, SRB measures as zero-noise limits, Ergod. Th. Dynamic. Systems, 25 (2005), 1115-1138.

[22]

L. J. Díaz, E. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43. doi: 10.1007/BF02392945.

[23]

T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions, Inventiones Math., 176 (2009), 617-636. doi: 10.1007/s00222-008-0172-4.

[24]

T. Downarowicz and S. Newhouse, Symbolic extensions in smooth dynamical systems, Inventiones Math., 160 (2005), 453-499. doi: 10.1007/s00222-004-0413-0.

[25]

N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations, Discrete Contin. Dyn. Syst., 26 (2010), 1-42. 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), 81-137. 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, Cambridge University Press, 1998.

[29]

R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8.

[30]

M. Misiurewicz, Diffeomorphim without any measure of maximal entropy, Bull. Acad. Pol. Sci., Ser Sci. Math, Astr. et Phys, 21 (1973), 903-910.

[31]

S. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc., 167 (1972), 125-150. doi: 10.1090/S0002-9947-1972-0295388-6.

[32]

S. Newhouse, Continuity properties of entropy, Annals of Math., 129 (1989), 215-235. 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), 293-317.

[35]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Math., 151 (2000), 961-1023. 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), 29-34. doi: 10.1016/j.topol.2009.04.053.

[37]

M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6.

[38]

M. Shub, "Global Stability of Dynamical Systems," Springer-Verlag, New York, 1987.

[39]

K. Sigmund, Generic properties of invariant measures for Axiom-A-diffeomorphisms, Inventiones Math., 11 (1970), 99-109. 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), 217-242.

[41]

P. Walters, "An Introduction to Ergodic Theory," volume 79 of Graduate Texts in Mathematics, Springer-Verlag, Berlin-New York, 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), 1-22.

[2]

M. Asaoka, Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions, Proc. Amer. Math. Soc., 136 (2008), 677-686. doi: 10.1090/S0002-9939-07-09115-0.

[3]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396. 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), 135-150.

[5]

C. Bonatti and S. Crovisier, Recurrence et généricité, Invent. Math., 158 (2004), 33-104.

[6]

C. Bonatti and L. J. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 171-197.

[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), 589-604.

[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), 355-418. 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 102 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2005.

[10]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193. 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), 187-222.

[12]

M. Boyle and T. Downarowicz, Symbolic extension entropy: $C^r$ examples, products and flows, Discrete Contin. Dyn. Syst., 16 (2006), 329-341. 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), 713-757. 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, (). 

[16]

K. Burns, personal, communication., (). 

[17]

J. Buzzi, Intrinsic ergodicity for smooth interval maps, Isreal J. Math., 100 (1997), 125-161. 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., (). 

[19]

A. Candel and L. Conlon, "Foliations I," volume 23 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2000.

[20]

C. Carbalo, C. Morales and M. J. Pacifico, Homoclinic classes for generic $C^1$ vector fields, Ergod. Th. Dynamic. Systems, 23 (2003), 403-415.

[21]

W. Cowieson and L.-S. Young, SRB measures as zero-noise limits, Ergod. Th. Dynamic. Systems, 25 (2005), 1115-1138.

[22]

L. J. Díaz, E. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43. doi: 10.1007/BF02392945.

[23]

T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions, Inventiones Math., 176 (2009), 617-636. doi: 10.1007/s00222-008-0172-4.

[24]

T. Downarowicz and S. Newhouse, Symbolic extensions in smooth dynamical systems, Inventiones Math., 160 (2005), 453-499. doi: 10.1007/s00222-004-0413-0.

[25]

N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations, Discrete Contin. Dyn. Syst., 26 (2010), 1-42. 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), 81-137. 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, Cambridge University Press, 1998.

[29]

R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8.

[30]

M. Misiurewicz, Diffeomorphim without any measure of maximal entropy, Bull. Acad. Pol. Sci., Ser Sci. Math, Astr. et Phys, 21 (1973), 903-910.

[31]

S. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc., 167 (1972), 125-150. doi: 10.1090/S0002-9947-1972-0295388-6.

[32]

S. Newhouse, Continuity properties of entropy, Annals of Math., 129 (1989), 215-235. 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), 293-317.

[35]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Math., 151 (2000), 961-1023. 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), 29-34. doi: 10.1016/j.topol.2009.04.053.

[37]

M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. doi: 10.1090/S0002-9904-1974-13344-6.

[38]

M. Shub, "Global Stability of Dynamical Systems," Springer-Verlag, New York, 1987.

[39]

K. Sigmund, Generic properties of invariant measures for Axiom-A-diffeomorphisms, Inventiones Math., 11 (1970), 99-109. 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), 217-242.

[41]

P. Walters, "An Introduction to Ergodic Theory," volume 79 of Graduate Texts in Mathematics, Springer-Verlag, Berlin-New York, 1982.

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