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

Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle

David Burguet 1, and  Todd Fisher 2,

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.
Keywords: Symbolic extensions, partial hyperbolicity..
Mathematics Subject Classification: 37C05, 37C40, 37A35, 37D30, 37B1.
Citation: David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

D. Burguet, $\mathcalC^2$ surface diffeomorphism have symbolic extensions, Invent. Math., 186 (2011), 191-236. 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), 357-369. doi: 10.1017/S0143385708080425.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Buzzi, Intrinsic ergodicity for smooth interval maps, Israel J. Math., 100 (1997), 125-161. 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), 1115-1138. 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), 1419-1441. 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, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155.  Google Scholar

[16]

T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116. 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), 617-636. 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), 453-499. 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, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235. 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), 197-231.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

D. Ruelle, An inequality of the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87. 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, Springer-Verlag, New York, 1987.  Google Scholar

[28]

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

[29]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. 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), 301-317. doi: 10.1007/BF02766216.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

D. Burguet, $\mathcalC^2$ surface diffeomorphism have symbolic extensions, Invent. Math., 186 (2011), 191-236. 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), 357-369. doi: 10.1017/S0143385708080425.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Buzzi, Intrinsic ergodicity for smooth interval maps, Israel J. Math., 100 (1997), 125-161. 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), 1115-1138. 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), 1419-1441. 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, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155.  Google Scholar

[16]

T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116. 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), 617-636. 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), 453-499. 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, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235. 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), 197-231.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

D. Ruelle, An inequality of the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87. 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, Springer-Verlag, New York, 1987.  Google Scholar

[28]

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

[29]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. 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), 301-317. doi: 10.1007/BF02766216.  Google Scholar

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