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July  2014, 34(7): 2819-2827. doi: 10.3934/dcds.2014.34.2819

Pointwise hyperbolicity implies uniform hyperbolicity

Boris Hasselblatt 1, , Yakov Pesin 2, and  Jörg Schmeling 3,

1. 

Department of Mathematics, Tufts University, Medford, MA 02155
2. 

Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802
3. 

Department of Mathematics, Lund Institute of Technology, Lunds Universitet, Box 118, SE-22100 Lund, Sweden

Received  March 2013 Revised  October 2013 Published  December 2013

We provide a general mechanism for obtaining uniform information from pointwise data. For instance, a diffeomorphism of a compact Riemannian manifold with pointwise expanding and contracting continuous invariant cone families is an Anosov diffeomorphism, i.e., the entire manifold is uniformly hyperbolic.
Keywords: nonuniform hyperbolicity, Uniform hyperbolicity, pointwise hyperbolicity..
Mathematics Subject Classification: Primary: 37D20, 37D25; Secondary: 54H0.
Citation: Boris Hasselblatt, Yakov Pesin, Jörg Schmeling. Pointwise hyperbolicity implies uniform hyperbolicity. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2819-2827. doi: 10.3934/dcds.2014.34.2819
References:
[1]

J. F. Alves, V. Araújo and B. Saussol, On the uniform hyperbolicity of some nonuniformly hyperbolic systems, Proc. Amer. Math. Soc., 131 (2003), 1303-1309. doi: 10.1090/S0002-9939-02-06857-0.  Google Scholar

[2]

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[3]

Y. Cao, Non-zero Lyapunov exponents and uniform hyperbolicity, Nonlinearity, 16 (2003), 1473-1479. doi: 10.1088/0951-7715/16/4/316.  Google Scholar

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Y. Cao, S. Luzzatto and I. Rios, A minimum principle for Lyapunov exponents and a higher-dimensional version of a theorem of Mañé, Qual. Theory Dyn. Syst., 5 (2004), 261-273. doi: 10.1007/BF02972681.  Google Scholar

[5]

Y. Cao, S. Luzzatto and I. Rios, Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies, Discrete Contin. Dyn. Syst., 15 (2006), 61-71. doi: 10.3934/dcds.2006.15.61.  Google Scholar

[6]

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

[7]

R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc., 229 (1977), 351-370.  Google Scholar

[8]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.  Google Scholar

show all references

References:
[1]

J. F. Alves, V. Araújo and B. Saussol, On the uniform hyperbolicity of some nonuniformly hyperbolic systems, Proc. Amer. Math. Soc., 131 (2003), 1303-1309. doi: 10.1090/S0002-9939-02-06857-0.  Google Scholar

[2]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, Cambridge, 2007.  Google Scholar

[3]

Y. Cao, Non-zero Lyapunov exponents and uniform hyperbolicity, Nonlinearity, 16 (2003), 1473-1479. doi: 10.1088/0951-7715/16/4/316.  Google Scholar

[4]

Y. Cao, S. Luzzatto and I. Rios, A minimum principle for Lyapunov exponents and a higher-dimensional version of a theorem of Mañé, Qual. Theory Dyn. Syst., 5 (2004), 261-273. doi: 10.1007/BF02972681.  Google Scholar

[5]

Y. Cao, S. Luzzatto and I. Rios, Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies, Discrete Contin. Dyn. Syst., 15 (2006), 61-71. doi: 10.3934/dcds.2006.15.61.  Google Scholar

[6]

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

[7]

R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc., 229 (1977), 351-370.  Google Scholar

[8]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.  Google Scholar

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