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

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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

Abstract
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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 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., ().
[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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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., ().
[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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