# American Institute of Mathematical Sciences

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). [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. [3] N. Bourbaki, General Topology. Chapters 1-4,, Reprint of the 1966 edition, (1966). [4] R. Bowen, Topological entropy for non-compact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. [5] M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions,, Invent. Math., 156 (2004), 119. doi: 10.1007/s00222-003-0335-2. [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. [7] J. Buzzi, Intrinsic ergodicity of smooth interval maps,, Israel J. Math., 100 (1997), 125. doi: 10.1007/BF02773637. [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. [9] J. Buzzi, A continuous, piecewise affine surface map with no measure of maximal entropy,, , (). [10] J. Buzzi, $C^r$ surface diffeomorphisms with no maximal entropy measure,, Erg. Th. Dynam. Syst., (). doi: 10.1017/etds.2013.25. [11] J. Buzzi, The almost Borel structure of diffeomorphisms with some hyperbolicity., Lecture notes. Hyperbolicity, (2013). [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. [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. [14] T. Downarowicz, Entropy structure,, J. Anal. Math., 96 (2005), 57. doi: 10.1007/BF02787825. [15] T. Downarowicz and S. Newhouse, Symbolic extensions in smooth dynamical systems,, Invent. Math., 160 (2005), 453. doi: 10.1007/s00222-004-0413-0. [16] T. Fisher, M. Sambarino and R. Potrie, Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov,, preprint, (). [17] M. Hochman, Isomorphism and embedding into Markov shifts off universally null sets,, Acta Applic. Math., 126 (2013), 187. [18] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. [19] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). [20] F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations,, J. London Math. Soc. (2), 16 (1977), 568. [21] R. Mañé, Ergodic Theory and Differentiable Dynamics,, Translated from the Portuguese by Silvio Levy, (1987). [22] M. Misiurewicz, Diffeomorphism without any measure with maximal entropy,, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903. [23] M. Misiurewicz, Topological conditional entropy,, Studia Math., 55 (1976), 175. [24] S. Newhouse and L.-S. Young, Dynamics of certain skew products,, in Geometric Dynamics (Rio de Janeiro, (1981), 611. doi: 10.1007/BFb0061436. [25] M. J. Pacifico and J. L. Vieitez, On measure expansive diffeomorphisms,, preprint, (). [26] J. Palis, Open questions leading to a global perspective in dynamics,, Nonlinearity, 21 (2008). doi: 10.1088/0951-7715/21/4/T01. [27] Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, Zurich Lectures in Advanced Mathematics, (2004). doi: 10.4171/003. [28] C. Robinson, Dynmical Systems Stability, Symbolic Dynamics, and Chaos,, Second edition, (1999). doi: 10.1117/12.343031. [29] F. Rodriguez Hertz, J. Rodriguez Hertz, A. Tahzibi and R. Ures, Maximizing measures for partially hyperbolic systems with compact center leaves,, preprint., (). [30] S. Ruette, Mixing Cr maps of the interval without maximal measure,, Israel J. Math., 127 (2002), 253. doi: 10.1007/BF02784534. [31] M. Shub, Topologically transitive diffeomorphisms on $T^4$,, in Dynamical Systems, (1971). [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. [33] B. Weiss, Intrinsically ergodic systems,, Bull. Amer. Math. Soc., 76 (1970), 1266. doi: 10.1090/S0002-9904-1970-12632-5.

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##### 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). [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. [3] N. Bourbaki, General Topology. Chapters 1-4,, Reprint of the 1966 edition, (1966). [4] R. Bowen, Topological entropy for non-compact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. [5] M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions,, Invent. Math., 156 (2004), 119. doi: 10.1007/s00222-003-0335-2. [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. [7] J. Buzzi, Intrinsic ergodicity of smooth interval maps,, Israel J. Math., 100 (1997), 125. doi: 10.1007/BF02773637. [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. [9] J. Buzzi, A continuous, piecewise affine surface map with no measure of maximal entropy,, , (). [10] J. Buzzi, $C^r$ surface diffeomorphisms with no maximal entropy measure,, Erg. Th. Dynam. Syst., (). doi: 10.1017/etds.2013.25. [11] J. Buzzi, The almost Borel structure of diffeomorphisms with some hyperbolicity., Lecture notes. Hyperbolicity, (2013). [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. [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. [14] T. Downarowicz, Entropy structure,, J. Anal. Math., 96 (2005), 57. doi: 10.1007/BF02787825. [15] T. Downarowicz and S. Newhouse, Symbolic extensions in smooth dynamical systems,, Invent. Math., 160 (2005), 453. doi: 10.1007/s00222-004-0413-0. [16] T. Fisher, M. Sambarino and R. Potrie, Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov,, preprint, (). [17] M. Hochman, Isomorphism and embedding into Markov shifts off universally null sets,, Acta Applic. Math., 126 (2013), 187. [18] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. [19] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). [20] F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations,, J. London Math. Soc. (2), 16 (1977), 568. [21] R. Mañé, Ergodic Theory and Differentiable Dynamics,, Translated from the Portuguese by Silvio Levy, (1987). [22] M. Misiurewicz, Diffeomorphism without any measure with maximal entropy,, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903. [23] M. Misiurewicz, Topological conditional entropy,, Studia Math., 55 (1976), 175. [24] S. Newhouse and L.-S. Young, Dynamics of certain skew products,, in Geometric Dynamics (Rio de Janeiro, (1981), 611. doi: 10.1007/BFb0061436. [25] M. J. Pacifico and J. L. Vieitez, On measure expansive diffeomorphisms,, preprint, (). [26] J. Palis, Open questions leading to a global perspective in dynamics,, Nonlinearity, 21 (2008). doi: 10.1088/0951-7715/21/4/T01. [27] Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, Zurich Lectures in Advanced Mathematics, (2004). doi: 10.4171/003. [28] C. Robinson, Dynmical Systems Stability, Symbolic Dynamics, and Chaos,, Second edition, (1999). doi: 10.1117/12.343031. [29] F. Rodriguez Hertz, J. Rodriguez Hertz, A. Tahzibi and R. Ures, Maximizing measures for partially hyperbolic systems with compact center leaves,, preprint., (). [30] S. Ruette, Mixing Cr maps of the interval without maximal measure,, Israel J. Math., 127 (2002), 253. doi: 10.1007/BF02784534. [31] M. Shub, Topologically transitive diffeomorphisms on $T^4$,, in Dynamical Systems, (1971). [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. [33] B. Weiss, Intrinsically ergodic systems,, Bull. Amer. Math. Soc., 76 (1970), 1266. doi: 10.1090/S0002-9904-1970-12632-5.
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