American Institute of Mathematical Sciences

Advanced
October  2013, 7(4): 527-552. doi: 10.3934/jmd.2013.7.527

Entropic stability beyond partial hyperbolicity

Jérôme Buzzi 1, and  Todd Fisher 2,

1. 

C.N.R.S. & Département de Mathématiques, Université Paris-Sud, 91405 Orsay, France
2. 

Department of Mathematics, Brigham Young University, Provo, UT 84602

Received  May 2012 Revised  August 2013 Published  March 2014

We analyze a class of $C^0$-small but $C^1$-large deformations of Anosov diffeomorphisms that break the topological conjugacy and structural stability, but unexpectedly retain the following stability property. The usual semiconjugacy mapping the deformation to the Anosov diffeomorphism is in fact an isomorphism with respect to all ergodic, invariant probability measures with entropy close to the maximum. In particular, the value of the topological entropy and the existence of a unique measure of maximal entropy are preserved. We also establish expansiveness around those measures. However, this expansivity is too weak to ensure the existence of symbolic extensions.
    Many constructions of robustly transitive diffeomorphisms can be done within this class. In particular, we show that it includes a class described by Bonatti and Viana of robustly transitive diffeomorphisms that are not partially hyperbolic.
Keywords: dominated splitting., ergodic theory, robust ergodicity, Measures of maximal entropy, topological entropy.
Mathematics Subject Classification: 37C40, 37A35, 37C1.
Citation: Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527
References:
[1]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005.  Google Scholar

[2]

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

[3]

N. Bourbaki, General Topology. Chapters 1-4, Reprint of the 1966 edition, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989.  Google Scholar

[4]

R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[5]

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

[6]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete Contin. Dyn. Syst., 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89.  Google Scholar

[7]

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

[8]

J. Buzzi, Dimenional entropies and semi-uniform hyperbolicity, in New Trends in Mathematical Physics. Selected Contributions of the XVth International Congress on Mathematical Physics (ed. V. Sidoravicius), Springer, 2009, 95-116. doi: 10.1007/978-90-481-2810-5_8.  Google Scholar

[9]

J. Buzzi, A continuous, piecewise affine surface map with no measure of maximal entropy,, , ().   Google Scholar

[10]

J. Buzzi, $C^r$ surface diffeomorphisms with no maximal entropy measure,, Erg. Th. Dynam. Syst., ().  doi: 10.1017/etds.2013.25.  Google Scholar

[11]

J. Buzzi, The almost Borel structure of diffeomorphisms with some hyperbolicity. Lecture notes. Hyperbolicity, large deviations and fluctuations, Lausanne 2013. Google Scholar

[12]

J. Buzzi, T. Fisher, M. Sambarino and C. Vásquez, Intrinsic ergodicity for certain nonhyperbolic robustly transitive systems, Erg. Th. Dynam. Syst., 32 (2012), 63-79. doi: 10.1017/S0143385710000854.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

T. Fisher, M. Sambarino and R. Potrie, Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov,, preprint, ().   Google Scholar

[17]

M. Hochman, Isomorphism and embedding into Markov shifts off universally null sets, Acta Applic. Math., 126 (2013), 187-201. Google Scholar

[18]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.  Google Scholar

[19]

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

[20]

F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc. (2), 16 (1977), 568-576.  Google Scholar

[21]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Translated from the Portuguese by Silvio Levy, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8, Springer-Verlag, Berlin, 1987.  Google Scholar

[22]

M. Misiurewicz, Diffeomorphism without any measure with maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903-910.  Google Scholar

[23]

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

[24]

S. Newhouse and L.-S. Young, Dynamics of certain skew products, in Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., 1007, Springer, Berlin, 1983, 611-629. doi: 10.1007/BFb0061436.  Google Scholar

[25]

M. J. Pacifico and J. L. Vieitez, On measure expansive diffeomorphisms,, preprint, ().   Google Scholar

[26]

J. Palis, Open questions leading to a global perspective in dynamics, Nonlinearity, 21 (2008), T37-T43. doi: 10.1088/0951-7715/21/4/T01.  Google Scholar

[27]

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/003.  Google Scholar

[28]

C. Robinson, Dynmical Systems Stability, Symbolic Dynamics, and Chaos, Second edition, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999. doi: 10.1117/12.343031.  Google Scholar

[29]

F. Rodriguez Hertz, J. Rodriguez Hertz, A. Tahzibi and R. Ures, Maximizing measures for partially hyperbolic systems with compact center leaves,, preprint., ().   Google Scholar

[30]

S. Ruette, Mixing Cr maps of the interval without maximal measure, Israel J. Math., 127 (2002), 253-277. doi: 10.1007/BF02784534.  Google Scholar

[31]

M. Shub, Topologically transitive diffeomorphisms on $T^4$, in Dynamical Systems, Lect. Notes in Math., 206, Springer Verlag, 1971, 39. Google Scholar

[32]

R. Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with hyperbolic linear part, Proc. Amer. Math. Soc., 140 (2012), 1973-1985. doi: 10.1090/S0002-9939-2011-11040-2.  Google Scholar

[33]

B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269. doi: 10.1090/S0002-9904-1970-12632-5.  Google Scholar

show all references

References:
[1]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005.  Google Scholar

[2]

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

[3]

N. Bourbaki, General Topology. Chapters 1-4, Reprint of the 1966 edition, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989.  Google Scholar

[4]

R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[5]

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

[6]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete Contin. Dyn. Syst., 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89.  Google Scholar

[7]

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

[8]

J. Buzzi, Dimenional entropies and semi-uniform hyperbolicity, in New Trends in Mathematical Physics. Selected Contributions of the XVth International Congress on Mathematical Physics (ed. V. Sidoravicius), Springer, 2009, 95-116. doi: 10.1007/978-90-481-2810-5_8.  Google Scholar

[9]

J. Buzzi, A continuous, piecewise affine surface map with no measure of maximal entropy,, , ().   Google Scholar

[10]

J. Buzzi, $C^r$ surface diffeomorphisms with no maximal entropy measure,, Erg. Th. Dynam. Syst., ().  doi: 10.1017/etds.2013.25.  Google Scholar

[11]

J. Buzzi, The almost Borel structure of diffeomorphisms with some hyperbolicity. Lecture notes. Hyperbolicity, large deviations and fluctuations, Lausanne 2013. Google Scholar

[12]

J. Buzzi, T. Fisher, M. Sambarino and C. Vásquez, Intrinsic ergodicity for certain nonhyperbolic robustly transitive systems, Erg. Th. Dynam. Syst., 32 (2012), 63-79. doi: 10.1017/S0143385710000854.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

T. Fisher, M. Sambarino and R. Potrie, Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov,, preprint, ().   Google Scholar

[17]

M. Hochman, Isomorphism and embedding into Markov shifts off universally null sets, Acta Applic. Math., 126 (2013), 187-201. Google Scholar

[18]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.  Google Scholar

[19]

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

[20]

F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc. (2), 16 (1977), 568-576.  Google Scholar

[21]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Translated from the Portuguese by Silvio Levy, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8, Springer-Verlag, Berlin, 1987.  Google Scholar

[22]

M. Misiurewicz, Diffeomorphism without any measure with maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903-910.  Google Scholar

[23]

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

[24]

S. Newhouse and L.-S. Young, Dynamics of certain skew products, in Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., 1007, Springer, Berlin, 1983, 611-629. doi: 10.1007/BFb0061436.  Google Scholar

[25]

M. J. Pacifico and J. L. Vieitez, On measure expansive diffeomorphisms,, preprint, ().   Google Scholar

[26]

J. Palis, Open questions leading to a global perspective in dynamics, Nonlinearity, 21 (2008), T37-T43. doi: 10.1088/0951-7715/21/4/T01.  Google Scholar

[27]

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/003.  Google Scholar

[28]

C. Robinson, Dynmical Systems Stability, Symbolic Dynamics, and Chaos, Second edition, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999. doi: 10.1117/12.343031.  Google Scholar

[29]

F. Rodriguez Hertz, J. Rodriguez Hertz, A. Tahzibi and R. Ures, Maximizing measures for partially hyperbolic systems with compact center leaves,, preprint., ().   Google Scholar

[30]

S. Ruette, Mixing Cr maps of the interval without maximal measure, Israel J. Math., 127 (2002), 253-277. doi: 10.1007/BF02784534.  Google Scholar

[31]

M. Shub, Topologically transitive diffeomorphisms on $T^4$, in Dynamical Systems, Lect. Notes in Math., 206, Springer Verlag, 1971, 39. Google Scholar

[32]

R. Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with hyperbolic linear part, Proc. Amer. Math. Soc., 140 (2012), 1973-1985. doi: 10.1090/S0002-9939-2011-11040-2.  Google Scholar

[33]

B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269. doi: 10.1090/S0002-9904-1970-12632-5.  Google Scholar

[1]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[2]

Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421
[3]

Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237
[4]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[5]

Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087
[6]

Wen Huang, Leiye Xu, Shengnan Xu. Ergodic measures of intermediate entropy for affine transformations of nilmanifolds. Electronic Research Archive, 2021, 29 (4) : 2819-2827. doi: 10.3934/era.2021015
[7]

Dante Carrasco-Olivera, Bernardo San Martín. Robust attractors without dominated splitting on manifolds with boundary. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4555-4563. doi: 10.3934/dcds.2014.34.4555
[8]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[9]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[10]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[11]

Xufeng Guo, Gang Liao, Wenxiang Sun, Dawei Yang. On the hybrid control of metric entropy for dominated splittings. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5011-5019. doi: 10.3934/dcds.2018219
[12]

Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192
[13]

Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235
[14]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201
[15]

Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827
[16]

Lluís Alsedà, David Juher, Francesc Mañosas. Forward triplets and topological entropy on trees. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021131
[17]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[18]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[19]

Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673
[20]

Domingo González, Gamaliel Blé. Core entropy of polynomials with a critical point of maximal order. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 115-130. doi: 10.3934/dcds.2019005

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (81)
  • HTML views (3)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences