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

Entropic stability beyond partial hyperbolicity

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.
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, (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. doi: 10.1007/BF02810585. Google Scholar

[3]

N. Bourbaki, General Topology. Chapters 1-4,, Reprint of the 1966 edition, (1966). Google Scholar

[4]

R. Bowen, Topological entropy for non-compact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. 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. 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. doi: 10.3934/dcds.2008.22.89. Google Scholar

[7]

J. Buzzi, Intrinsic ergodicity of smooth interval maps,, Israel J. Math., 100 (1997), 125. 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), (2009), 95. 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, (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. 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. doi: 10.3934/dcds.2011.29.1419. Google Scholar

[14]

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

[15]

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

[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. Google Scholar

[18]

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

[19]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). Google Scholar

[20]

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

[21]

R. Mañé, Ergodic Theory and Differentiable Dynamics,, Translated from the Portuguese by Silvio Levy, (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. Google Scholar

[23]

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

[24]

S. Newhouse and L.-S. Young, Dynamics of certain skew products,, in Geometric Dynamics (Rio de Janeiro, (1981), 611. 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). 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, (2004). doi: 10.4171/003. Google Scholar

[28]

C. Robinson, Dynmical Systems Stability, Symbolic Dynamics, and Chaos,, Second edition, (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. doi: 10.1007/BF02784534. Google Scholar

[31]

M. Shub, Topologically transitive diffeomorphisms on $T^4$,, in Dynamical Systems, (1971). Google Scholar

[32]

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

[33]

B. Weiss, Intrinsically ergodic systems,, Bull. Amer. Math. Soc., 76 (1970), 1266. 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, (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. doi: 10.1007/BF02810585. Google Scholar

[3]

N. Bourbaki, General Topology. Chapters 1-4,, Reprint of the 1966 edition, (1966). Google Scholar

[4]

R. Bowen, Topological entropy for non-compact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. 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. 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. doi: 10.3934/dcds.2008.22.89. Google Scholar

[7]

J. Buzzi, Intrinsic ergodicity of smooth interval maps,, Israel J. Math., 100 (1997), 125. 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), (2009), 95. 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, (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. 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. doi: 10.3934/dcds.2011.29.1419. Google Scholar

[14]

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

[15]

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

[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. Google Scholar

[18]

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

[19]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). Google Scholar

[20]

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

[21]

R. Mañé, Ergodic Theory and Differentiable Dynamics,, Translated from the Portuguese by Silvio Levy, (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. Google Scholar

[23]

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

[24]

S. Newhouse and L.-S. Young, Dynamics of certain skew products,, in Geometric Dynamics (Rio de Janeiro, (1981), 611. 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). 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, (2004). doi: 10.4171/003. Google Scholar

[28]

C. Robinson, Dynmical Systems Stability, Symbolic Dynamics, and Chaos,, Second edition, (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. doi: 10.1007/BF02784534. Google Scholar

[31]

M. Shub, Topologically transitive diffeomorphisms on $T^4$,, in Dynamical Systems, (1971). Google Scholar

[32]

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

[33]

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

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