# American Institute of Mathematical Sciences

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

## 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

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.
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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