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A generic-dimensional property of the invariant measures for circle diffeomorphisms
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Bowen's construction for the Teichmüller flow
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 |
  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.
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. |
[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. |
[3] |
N. Bourbaki, General Topology. Chapters 1-4, Reprint of the 1966 edition, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. |
[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. |
[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. |
[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. |
[7] |
J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161.
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), Springer, 2009, 95-116.
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, large deviations and fluctuations, Lausanne 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), 63-79.
doi: 10.1017/S0143385710000854. |
[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. |
[14] |
T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116.
doi: 10.1007/BF02787825. |
[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. |
[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), 187-201. |
[18] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. |
[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. |
[20] |
F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc. (2), 16 (1977), 568-576. |
[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. |
[22] |
M. Misiurewicz, Diffeomorphism without any measure with maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903-910. |
[23] |
M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200. |
[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. |
[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), T37-T43.
doi: 10.1088/0951-7715/21/4/T01. |
[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. |
[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. |
[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), 253-277.
doi: 10.1007/BF02784534. |
[31] |
M. Shub, Topologically transitive diffeomorphisms on $T^4$, in Dynamical Systems, Lect. Notes in Math., 206, Springer Verlag, 1971, 39. |
[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. |
[33] |
B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269.
doi: 10.1090/S0002-9904-1970-12632-5. |
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. |
[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. |
[3] |
N. Bourbaki, General Topology. Chapters 1-4, Reprint of the 1966 edition, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. |
[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. |
[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. |
[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. |
[7] |
J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161.
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), Springer, 2009, 95-116.
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, large deviations and fluctuations, Lausanne 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), 63-79.
doi: 10.1017/S0143385710000854. |
[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. |
[14] |
T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116.
doi: 10.1007/BF02787825. |
[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. |
[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), 187-201. |
[18] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. |
[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. |
[20] |
F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc. (2), 16 (1977), 568-576. |
[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. |
[22] |
M. Misiurewicz, Diffeomorphism without any measure with maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903-910. |
[23] |
M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200. |
[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. |
[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), T37-T43.
doi: 10.1088/0951-7715/21/4/T01. |
[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. |
[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. |
[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), 253-277.
doi: 10.1007/BF02784534. |
[31] |
M. Shub, Topologically transitive diffeomorphisms on $T^4$, in Dynamical Systems, Lect. Notes in Math., 206, Springer Verlag, 1971, 39. |
[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. |
[33] |
B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc., 76 (1970), 1266-1269.
doi: 10.1090/S0002-9904-1970-12632-5. |
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