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 and Continuous Dynamical Systems, 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), 677-686. doi: 10.1090/S0002-9939-07-09115-0.

[2]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps, Ann. of Math. (2), 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423.

[3]

R. Bowen, Entropy-expansive maps, Trans. Ame. Math. Soc., 164 (1972), 323-331.

[4]

M. Boyle and T. Downarowicz, The entropy theory of symbolic extension, Invent. Math., 156 (2004), 119-161 . doi: 10.1007/s00222-003-0335-2.

[5]

M. Boyle and T. Downarowicz, Symbolic extension entropy : $\mathcalC^r$ examples, products and flows, Discrete Contin. Dyn. Syst., 16 (2006), 329-341. doi: 10.3934/dcds.2006.16.329.

[6]

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.

[7]

D. Burguet, $\mathcalC^2$ surface diffeomorphism have symbolic extensions, Invent. Math., 186 (2011), 191-236. doi: 10.1007/s00222-011-0317-8.

[8]

D. Burguet, A direct proof of the variational principle for tail entropy and its extension to maps, Ergodic Theory Dynam. Systems, 29 (2009), 357-369. doi: 10.1017/S0143385708080425.

[9]

D. Burguet, Symbolic extension for $\mathcalC^r$ non uniformly entropy expanding maps, Colloq. Math., 121 (2010), 129-151. doi: 10.4064/cm121-1-12.

[10]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451.

[11]

J. Buzzi, Intrinsic ergodicity for smooth interval maps, Israel J. Math., 100 (1997), 125-161. doi: 10.1007/BF02773637.

[12]

W. Cowieson and L.-S. Young, SRB mesaures as zero-noise limits, Ergod. Th. Dynamic. Systems, 25 (2005), 1115-1138. doi: 10.1017/S0143385704000604.

[13]

L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 29 (2011), 1419-1441. doi: 10.3934/dcds.2011.29.1419.

[14]

L. J. Diaz, T. Fisher, M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., to appear, arXiv:1010.0721.

[15]

T. Downarowicz, "Entropy in Dynamical Systems, New Mathematical Monographs," 18, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155.

[16]

T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116. doi: 10.1007/BF02787825.

[17]

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

[18]

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

[19]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes In Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.

[20]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and Its Applications, 54, Cambridge University Press, Cambridge, 1995.

[21]

M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.

[22]

S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235. doi: 10.2307/1971492.

[23]

V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 197-231.

[24]

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

[25]

Y. Pesin and L. Barreira, "Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Mathematics and Its Applications, 115, Cambridge University Press, Cambridge, 2007.

[26]

D. Ruelle, An inequality of the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87. doi: 10.1007/BF02584795.

[27]

M. Shub, "Global Stability of Dynamical Systems," With the collaboration of A. Fathi and R. Langevin. Transl. by J. Cristy, Springer-Verlag, New York, 1987.

[28]

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

[29]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.

[30]

Y. Yomdin, $\mathcalC^k$-resolution of semialgebraic mappings. Addendum to : "Volume growth and entropy", Israel J. Math., 57 (1987), 301-317. doi: 10.1007/BF02766216.

show all references

References:
[1]

M. Asaoka, Hyperbolic set exhibing $\mathcalC^1$-persistent homoclinic tangency for higher dimensions, Proc. Am. Math. Soc., 136 (2008), 677-686. doi: 10.1090/S0002-9939-07-09115-0.

[2]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps, Ann. of Math. (2), 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423.

[3]

R. Bowen, Entropy-expansive maps, Trans. Ame. Math. Soc., 164 (1972), 323-331.

[4]

M. Boyle and T. Downarowicz, The entropy theory of symbolic extension, Invent. Math., 156 (2004), 119-161 . doi: 10.1007/s00222-003-0335-2.

[5]

M. Boyle and T. Downarowicz, Symbolic extension entropy : $\mathcalC^r$ examples, products and flows, Discrete Contin. Dyn. Syst., 16 (2006), 329-341. doi: 10.3934/dcds.2006.16.329.

[6]

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.

[7]

D. Burguet, $\mathcalC^2$ surface diffeomorphism have symbolic extensions, Invent. Math., 186 (2011), 191-236. doi: 10.1007/s00222-011-0317-8.

[8]

D. Burguet, A direct proof of the variational principle for tail entropy and its extension to maps, Ergodic Theory Dynam. Systems, 29 (2009), 357-369. doi: 10.1017/S0143385708080425.

[9]

D. Burguet, Symbolic extension for $\mathcalC^r$ non uniformly entropy expanding maps, Colloq. Math., 121 (2010), 129-151. doi: 10.4064/cm121-1-12.

[10]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451.

[11]

J. Buzzi, Intrinsic ergodicity for smooth interval maps, Israel J. Math., 100 (1997), 125-161. doi: 10.1007/BF02773637.

[12]

W. Cowieson and L.-S. Young, SRB mesaures as zero-noise limits, Ergod. Th. Dynamic. Systems, 25 (2005), 1115-1138. doi: 10.1017/S0143385704000604.

[13]

L. J. Díaz and T. Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 29 (2011), 1419-1441. doi: 10.3934/dcds.2011.29.1419.

[14]

L. J. Diaz, T. Fisher, M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., to appear, arXiv:1010.0721.

[15]

T. Downarowicz, "Entropy in Dynamical Systems, New Mathematical Monographs," 18, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155.

[16]

T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116. doi: 10.1007/BF02787825.

[17]

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

[18]

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

[19]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes In Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.

[20]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and Its Applications, 54, Cambridge University Press, Cambridge, 1995.

[21]

M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.

[22]

S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235. doi: 10.2307/1971492.

[23]

V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 197-231.

[24]

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

[25]

Y. Pesin and L. Barreira, "Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Mathematics and Its Applications, 115, Cambridge University Press, Cambridge, 2007.

[26]

D. Ruelle, An inequality of the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87. doi: 10.1007/BF02584795.

[27]

M. Shub, "Global Stability of Dynamical Systems," With the collaboration of A. Fathi and R. Langevin. Transl. by J. Cristy, Springer-Verlag, New York, 1987.

[28]

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

[29]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.

[30]

Y. Yomdin, $\mathcalC^k$-resolution of semialgebraic mappings. Addendum to : "Volume growth and entropy", Israel J. Math., 57 (1987), 301-317. doi: 10.1007/BF02766216.

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