# American Institute of Mathematical Sciences

June  2013, 33(6): 2253-2270. doi: 10.3934/dcds.2013.33.2253

## 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

Received  July 2011 Revised  July 2012 Published  December 2012

We relate the symbolic extension entropy of a partially hyperbolic dynamical system to the entropy appearing at small scales in local center manifolds. In particular, we prove the existence of symbolic extensions for $\mathcal{C}^2$ partially hyperbolic diffeomorphisms with a $2$-dimensional center bundle. 200 words.
Citation: David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253
##### References:
 [1] M. Asaoka, Hyperbolic set exhibing $\mathcalC^1$-persistent homoclinic tangency for higher dimensions,, Proc. Am. Math. Soc., 136 (2008), 677. doi: 10.1090/S0002-9939-07-09115-0. Google Scholar [2] J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps,, Ann. of Math. (2), 161 (2005), 1423. doi: 10.4007/annals.2005.161.1423. Google Scholar [3] R. Bowen, Entropy-expansive maps,, Trans. Ame. Math. Soc., 164 (1972), 323. Google Scholar [4] M. Boyle and T. Downarowicz, The entropy theory of symbolic extension,, Invent. Math., 156 (2004), 119. doi: 10.1007/s00222-003-0335-2. Google Scholar [5] M. Boyle and T. Downarowicz, Symbolic extension entropy : $\mathcalC^r$ examples, products and flows,, Discrete Contin. Dyn. Syst., 16 (2006), 329. doi: 10.3934/dcds.2006.16.329. Google Scholar [6] 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. Google Scholar [7] D. Burguet, $\mathcalC^2$ surface diffeomorphism have symbolic extensions,, Invent. Math., 186 (2011), 191. doi: 10.1007/s00222-011-0317-8. Google Scholar [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. doi: 10.1017/S0143385708080425. Google Scholar [9] D. Burguet, Symbolic extension for $\mathcalC^r$ non uniformly entropy expanding maps,, Colloq. Math., 121 (2010), 129. doi: 10.4064/cm121-1-12. Google Scholar [10] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems,, Ann. of Math. (2), 171 (2010), 451. doi: 10.4007/annals.2010.171.451. Google Scholar [11] J. Buzzi, Intrinsic ergodicity for smooth interval maps,, Israel J. Math., 100 (1997), 125. doi: 10.1007/BF02773637. Google Scholar [12] W. Cowieson and L.-S. Young, SRB mesaures as zero-noise limits,, Ergod. Th. Dynamic. Systems, 25 (2005), 1115. doi: 10.1017/S0143385704000604. Google Scholar [13] L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms,, Discrete Contin. Dyn. Syst., 29 (2011), 1419. doi: 10.3934/dcds.2011.29.1419. Google Scholar [14] L. J. Diaz, T. Fisher, M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms,, Discrete Contin. Dyn. Syst., (). Google Scholar [15] T. Downarowicz, "Entropy in Dynamical Systems, New Mathematical Monographs,", 18, 18 (2011). doi: 10.1017/CBO9780511976155. Google Scholar [16] T. Downarowicz, Entropy structure,, J. Anal. Math., 96 (2005), 57. doi: 10.1007/BF02787825. Google Scholar [17] T. Downarowicz and A. Maass, Smooth interval maps have symbolic extensions: the antarctic theorem,, Invent. Math., 176 (2009), 617. doi: 10.1007/s00222-008-0172-4. Google Scholar [18] T. Downarowicz and S. Newhouse, Symbolic extensions in smooth dynamical systems,, Invent. Math., 160 (2005), 453. doi: 10.1007/s00222-004-0413-0. Google Scholar [19] M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes In Mathematics, 583 (1977). Google Scholar [20] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Mathematics and Its Applications, 54 (1995). Google Scholar [21] M. Misiurewicz, Topological conditional entropy,, Studia Math., 55 (1976), 175. Google Scholar [22] S. Newhouse, Continuity properties of entropy,, Ann. of Math. (2), 129 (1989), 215. doi: 10.2307/1971492. Google Scholar [23] V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems,, Trudy Moskov. Mat. Obšč., 19 (1968), 197. Google Scholar [24] M. Pacifico and J. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms,, Rev. Mat. Complut., 21 (2008), 293. Google Scholar [25] Y. Pesin and L. Barreira, "Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents,", Encyclopedia of Mathematics and Its Applications, 115 (2007). Google Scholar [26] D. Ruelle, An inequality of the entropy of differentiable maps,, Bol. Soc. Brasil. Mat., 9 (1978), 83. doi: 10.1007/BF02584795. Google Scholar [27] M. Shub, "Global Stability of Dynamical Systems,", With the collaboration of A. Fathi and R. Langevin. Transl. by J. Cristy, (1987). Google Scholar [28] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982). Google Scholar [29] Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215. Google Scholar [30] Y. Yomdin, $\mathcalC^k$-resolution of semialgebraic mappings. Addendum to : "Volume growth and entropy",, Israel J. Math., 57 (1987), 301. doi: 10.1007/BF02766216. Google Scholar

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