American Institute of Mathematical Sciences

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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396. doi: 10.2307/2118647.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Ch. Bonatti, L. J. Díaz, E.R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles, Astérisque, 286 (2003), 187-222.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Brin and Stuck G, "Introduction to Dynamical Systems," Cambridge University Press, 2002. doi: 10.1017/CBO9780511755316.  Google Scholar

[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), 125-161. doi: 10.1007/BF02773637.  Google Scholar

[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 23 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, 1995.  Google Scholar

[28]

G. Keller, "Equilibrium States in Ergodic Theory," London Mathematical Society Student Texts, Cambridge University Press, 1998.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

S. Newhouse, Continuity properties of entropy, Annals of Math., 129 (1989), 215-235. doi: 10.2307/1971492.  Google Scholar

[33]

M. J. Pacifico and J. Vieitez, Robust entropy-expansiveness implies generic domination,, preprint, ().   Google Scholar

[34]

M. J. Pacifico and J. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317.  Google Scholar

[35]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Math., 151 (2000), 961-1023. doi: 10.2307/121127.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[39]

K. Sigmund, Generic properties of invariant measures for Axiom-A-diffeomorphisms, Inventiones Math., 11 (1970), 99-109. doi: 10.1007/BF01404606.  Google Scholar

[40]

C. P. Simon, Instability in Diff$(T^3)$ and the nongenericity of rational zeta function, Trans. Amer. Math. Soc., 174 (1972), 217-242.  Google Scholar

[41]

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

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.  Google Scholar

[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.  Google Scholar

[3]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396. doi: 10.2307/2118647.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Ch. Bonatti, L. J. Díaz, E.R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles, Astérisque, 286 (2003), 187-222.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Brin and Stuck G, "Introduction to Dynamical Systems," Cambridge University Press, 2002. doi: 10.1017/CBO9780511755316.  Google Scholar

[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), 125-161. doi: 10.1007/BF02773637.  Google Scholar

[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 23 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, 1995.  Google Scholar

[28]

G. Keller, "Equilibrium States in Ergodic Theory," London Mathematical Society Student Texts, Cambridge University Press, 1998.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

S. Newhouse, Continuity properties of entropy, Annals of Math., 129 (1989), 215-235. doi: 10.2307/1971492.  Google Scholar

[33]

M. J. Pacifico and J. Vieitez, Robust entropy-expansiveness implies generic domination,, preprint, ().   Google Scholar

[34]

M. J. Pacifico and J. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317.  Google Scholar

[35]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Math., 151 (2000), 961-1023. doi: 10.2307/121127.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[39]

K. Sigmund, Generic properties of invariant measures for Axiom-A-diffeomorphisms, Inventiones Math., 11 (1970), 99-109. doi: 10.1007/BF01404606.  Google Scholar

[40]

C. P. Simon, Instability in Diff$(T^3)$ and the nongenericity of rational zeta function, Trans. Amer. Math. Soc., 174 (1972), 217-242.  Google Scholar

[41]

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

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