December  2012, 32(12): 4287-4305. doi: 10.3934/dcds.2012.32.4287

Some results on perturbations of Lyapunov exponents

1. 

Applied Mathematical Department, Central University of Finance and Economics, Beijing, 100081, China

2. 

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871

3. 

Departamento de Geometria, Instituto de Matemática, Universidade Federal Fluminense, Niterói, 24020-140, Brazil

Received  November 2010 Revised  May 2012 Published  August 2012

In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we can perturb all the zero integrated Lyapunov exponents $\int_M \lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among the conservative diffeomorphisms far from tangencies, we prove that ergodic ones form a residual subset.
Citation: Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287
References:
[1]

F. Abdenur and C. Bonatti, S. Crovisier, L. J. Díaz and L. Wen, Periodic points and homoclinic classes,, Ergod. Th. Dynam. Sys., 27 (2007), 1.  doi: 10.1017/S0143385706000538.  Google Scholar

[2]

A. Arbieto and T. Catalan, Hyperbolicity in the volume preserving scenario,, preprint, ().   Google Scholar

[3]

M. C. Arnaud, Cr$\acutee$ation de connexions en topologie $C^1$,, (French) [Creating connections in the $C^1$-topology], 21 (2001), 339.   Google Scholar

[4]

L. Arnold, "Random Dynamical Systems,", Springer-Verlag, (1998).   Google Scholar

[5]

L. Arnold and N. D. Cong, On the simplicity of the Lyapunov spectrum ofproducts of random matrices,, Ergod. Th. Dynam. Sys., 17 (1997), 1005.  doi: 10.1017/S0143385797086355.  Google Scholar

[6]

L. Arnold and N. D. Cong, Linear cocycles with simple Lyapunov spectrum are dense in $L^\infty$,, Ergod. Th. Dynam. Sys., 19 (1999), 1389.  doi: 10.1017/S014338579915199X.  Google Scholar

[7]

A. Ávila, On the regularization of conservative maps,, Acta Mathematica, 205 (2010), 5.   Google Scholar

[8]

A. Ávila, J. Bochi and A. Wilkinson, Nonuniform center bunching andthe genericity of ergodicity among $C^1$ partially hyperbolicsymplectomotphisms,, Annales Scientifiques de l'Ecole Normale Superieure, 42 (2009), 931.   Google Scholar

[9]

A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents,, Ergod. Th. Dynam. Sys., 22 (2003), 1655.  doi: 10.1017/S0143385702001773.  Google Scholar

[10]

L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", A. M. S., (2002).   Google Scholar

[11]

J. Bochi, Genericity of zero Lyapunov exponents,, Ergod. Th. Dynam. Sys., 22 (2002), 1667.  doi: 10.1017/S0143385702001165.  Google Scholar

[12]

J. Bochi, B. R. Fayad and E. Pujals, A remark on conservative diffeomorphisms,, C.R.Math. Acad. Sci. Paris, 342 (2006), 763.   Google Scholar

[13]

J. Bochi and M. Viana, The Lyapunov exponents of genericvolume-preserving and symplectic maps,, Ann. of Math., 161 (2005), 1423.  doi: 10.4007/annals.2005.161.1423.  Google Scholar

[14]

C. Bonatti and S. Crovisier, Récurrence et généricité,, (French) [Recurrence and genericity], 158 (2004), 33.   Google Scholar

[15]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms,, Ann. of Math., 143 (1996), 357.  doi: 10.2307/2118647.  Google Scholar

[16]

C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics,, Journal of the Inst. of Math. Jussieu, 7 (2008), 469.   Google Scholar

[17]

C. Bonatti, L. J. Díaz and E.R. Pujals, A $C^1-$generic dichotomy for diffeomorphisms: Weak forms ofhyperbolicity or infinitely many sinks or sources,, Ann. of Math., 158 (2003), 355.  doi: 10.4007/annals.2003.158.355.  Google Scholar

[18]

C. Bonatti, L. J. Díaz, E.R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles,, Astérisque, 286 (2003), 187.   Google Scholar

[19]

C. Bonatti, L. J. Díaz and M. Viana, Discontinuity of the Hausdorff dimension of hyperbolic sets,, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 713.   Google Scholar

[20]

C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity:A Global Geometric and Probabilistic Perspective,", Encyclopaedia of MathematicalSciences, 102 (2005).   Google Scholar

[21]

C. Bonatti and M. Viana, Lyapunov exponents with multiplicity 1 fordeterministic products of matrices,, Ergod. Th. Dynam. Sys., 24 (2004), 1295.  doi: 10.1017/S0143385703000695.  Google Scholar

[22]

K. Burns, D. Dolgopyat and Y. Pesin, Partial hyperbolicity, Lyapunovexponents and stable ergodicity,, J. Stat. Phys., 109 (2002), 927.  doi: 10.1023/A:1019779128351.  Google Scholar

[23]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic sysytems,, Ann. of Math., 171 (2010), 451.  doi: 10.4007/annals.2010.171.451.  Google Scholar

[24]

L. J. Díaz, Robust nonhyperbolic dynamics and heterodimensional cycles,, Ergod. Th. Dynam. Sys., 15 (1995), 291.   Google Scholar

[25]

D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense,, Geometric Methods in Dynamics, 287 (2003), 33.   Google Scholar

[26]

S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega-$stable conjecture for flows,, Ann. of Math., 145 (1997), 81.   Google Scholar

[27]

F. R. Hertz, M. A. R. Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and non-uniform hyperbolicity,, Duke Math. J., 160 (2011), 599.  doi: 10.1215/00127094-1444314.  Google Scholar

[28]

F. R. Hertz, M. A. R. Hertz, A. Tahzibi and R. Ures, Creation of Blenders in the conservative setting,, Nonlinearity (Bristol), 23 (2010), 211.   Google Scholar

[29]

O. Knill, Positive Lyapunov indices determine absolutely continuousspectra of stationary random one-dimensional Schrodinger operators,, Stochastic analysis, (1984), 225.   Google Scholar

[30]

C. Liang and G. Liu, Dominated splitting versus Small angles,, Acta Math. Sinica, 24 (2008), 1163.  doi: 10.1007/s10114-007-6445-9.  Google Scholar

[31]

C. Liang and G. Liu, Conditions for dominated splitting,, Acta Math. Sinica, 25 (2009), 1389.  doi: 10.1007/s10114-009-6568-2.  Google Scholar

[32]

C. Liang, G. Liu and W. Sun, Equivalent conditions fordominated splitting in volume-preserving diffeomorphisms,, Acta Math. Sinica, 23 (2007), 1563.  doi: 10.1007/s10114-005-0889-6.  Google Scholar

[33]

R. Mañé, An ergodic closing lemma,, Ann. of Math., 116 (1982), 503.  doi: 10.2307/2007021.  Google Scholar

[34]

M. Pollicott, "Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds,", Cambridge Univ. Press, (1993).   Google Scholar

[35]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Ann. of Math. 151 (2000), 151 (2000), 961.   Google Scholar

[36]

M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents,, Invent. Math., 139 (2000), 495.  doi: 10.1007/s002229900035.  Google Scholar

[37]

P. Walters, "An Introduction to Ergodic Theory,", Springer, (1982).   Google Scholar

[38]

Z. Wang and W. Sun, Lyapunov exponents of hyperbolic measures andhyperbolic periodic orbits,, Trans. Amer. Math., 362 (2010), 4267.  doi: 10.1090/S0002-9947-10-04947-0.  Google Scholar

[39]

L. Wen, Homoclinic tangengcies and dominated splittings,, Nonlinearity, 15 (2002), 1445.  doi: 10.1088/0951-7715/15/5/306.  Google Scholar

[40]

L. Wen, A uniform $C^1$ connecting lemma,, Discrete and Continuous Dynamical Systems, 8 (2002), 257.   Google Scholar

[41]

L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensionalcycles,, Bull. Braz. Math. Soc., (2004), 419.  doi: 10.1007/s00574-004-0023-x.  Google Scholar

[42]

L. Wen and Z. Xia, $C^1$ connecting lemmas,, Trans. Amer. Math., 352 (2000), 5213.  doi: 10.1090/S0002-9947-00-02553-8.  Google Scholar

[43]

J. Yang, "$C^1$ Dynamics Far From Tangencies,", Ph.D thesis, (2008).   Google Scholar

[44]

J. Yang, Ergodic measures far away from tangencies,, preprint, (2009).   Google Scholar

show all references

References:
[1]

F. Abdenur and C. Bonatti, S. Crovisier, L. J. Díaz and L. Wen, Periodic points and homoclinic classes,, Ergod. Th. Dynam. Sys., 27 (2007), 1.  doi: 10.1017/S0143385706000538.  Google Scholar

[2]

A. Arbieto and T. Catalan, Hyperbolicity in the volume preserving scenario,, preprint, ().   Google Scholar

[3]

M. C. Arnaud, Cr$\acutee$ation de connexions en topologie $C^1$,, (French) [Creating connections in the $C^1$-topology], 21 (2001), 339.   Google Scholar

[4]

L. Arnold, "Random Dynamical Systems,", Springer-Verlag, (1998).   Google Scholar

[5]

L. Arnold and N. D. Cong, On the simplicity of the Lyapunov spectrum ofproducts of random matrices,, Ergod. Th. Dynam. Sys., 17 (1997), 1005.  doi: 10.1017/S0143385797086355.  Google Scholar

[6]

L. Arnold and N. D. Cong, Linear cocycles with simple Lyapunov spectrum are dense in $L^\infty$,, Ergod. Th. Dynam. Sys., 19 (1999), 1389.  doi: 10.1017/S014338579915199X.  Google Scholar

[7]

A. Ávila, On the regularization of conservative maps,, Acta Mathematica, 205 (2010), 5.   Google Scholar

[8]

A. Ávila, J. Bochi and A. Wilkinson, Nonuniform center bunching andthe genericity of ergodicity among $C^1$ partially hyperbolicsymplectomotphisms,, Annales Scientifiques de l'Ecole Normale Superieure, 42 (2009), 931.   Google Scholar

[9]

A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents,, Ergod. Th. Dynam. Sys., 22 (2003), 1655.  doi: 10.1017/S0143385702001773.  Google Scholar

[10]

L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", A. M. S., (2002).   Google Scholar

[11]

J. Bochi, Genericity of zero Lyapunov exponents,, Ergod. Th. Dynam. Sys., 22 (2002), 1667.  doi: 10.1017/S0143385702001165.  Google Scholar

[12]

J. Bochi, B. R. Fayad and E. Pujals, A remark on conservative diffeomorphisms,, C.R.Math. Acad. Sci. Paris, 342 (2006), 763.   Google Scholar

[13]

J. Bochi and M. Viana, The Lyapunov exponents of genericvolume-preserving and symplectic maps,, Ann. of Math., 161 (2005), 1423.  doi: 10.4007/annals.2005.161.1423.  Google Scholar

[14]

C. Bonatti and S. Crovisier, Récurrence et généricité,, (French) [Recurrence and genericity], 158 (2004), 33.   Google Scholar

[15]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms,, Ann. of Math., 143 (1996), 357.  doi: 10.2307/2118647.  Google Scholar

[16]

C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics,, Journal of the Inst. of Math. Jussieu, 7 (2008), 469.   Google Scholar

[17]

C. Bonatti, L. J. Díaz and E.R. Pujals, A $C^1-$generic dichotomy for diffeomorphisms: Weak forms ofhyperbolicity or infinitely many sinks or sources,, Ann. of Math., 158 (2003), 355.  doi: 10.4007/annals.2003.158.355.  Google Scholar

[18]

C. Bonatti, L. J. Díaz, E.R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles,, Astérisque, 286 (2003), 187.   Google Scholar

[19]

C. Bonatti, L. J. Díaz and M. Viana, Discontinuity of the Hausdorff dimension of hyperbolic sets,, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 713.   Google Scholar

[20]

C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity:A Global Geometric and Probabilistic Perspective,", Encyclopaedia of MathematicalSciences, 102 (2005).   Google Scholar

[21]

C. Bonatti and M. Viana, Lyapunov exponents with multiplicity 1 fordeterministic products of matrices,, Ergod. Th. Dynam. Sys., 24 (2004), 1295.  doi: 10.1017/S0143385703000695.  Google Scholar

[22]

K. Burns, D. Dolgopyat and Y. Pesin, Partial hyperbolicity, Lyapunovexponents and stable ergodicity,, J. Stat. Phys., 109 (2002), 927.  doi: 10.1023/A:1019779128351.  Google Scholar

[23]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic sysytems,, Ann. of Math., 171 (2010), 451.  doi: 10.4007/annals.2010.171.451.  Google Scholar

[24]

L. J. Díaz, Robust nonhyperbolic dynamics and heterodimensional cycles,, Ergod. Th. Dynam. Sys., 15 (1995), 291.   Google Scholar

[25]

D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense,, Geometric Methods in Dynamics, 287 (2003), 33.   Google Scholar

[26]

S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega-$stable conjecture for flows,, Ann. of Math., 145 (1997), 81.   Google Scholar

[27]

F. R. Hertz, M. A. R. Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and non-uniform hyperbolicity,, Duke Math. J., 160 (2011), 599.  doi: 10.1215/00127094-1444314.  Google Scholar

[28]

F. R. Hertz, M. A. R. Hertz, A. Tahzibi and R. Ures, Creation of Blenders in the conservative setting,, Nonlinearity (Bristol), 23 (2010), 211.   Google Scholar

[29]

O. Knill, Positive Lyapunov indices determine absolutely continuousspectra of stationary random one-dimensional Schrodinger operators,, Stochastic analysis, (1984), 225.   Google Scholar

[30]

C. Liang and G. Liu, Dominated splitting versus Small angles,, Acta Math. Sinica, 24 (2008), 1163.  doi: 10.1007/s10114-007-6445-9.  Google Scholar

[31]

C. Liang and G. Liu, Conditions for dominated splitting,, Acta Math. Sinica, 25 (2009), 1389.  doi: 10.1007/s10114-009-6568-2.  Google Scholar

[32]

C. Liang, G. Liu and W. Sun, Equivalent conditions fordominated splitting in volume-preserving diffeomorphisms,, Acta Math. Sinica, 23 (2007), 1563.  doi: 10.1007/s10114-005-0889-6.  Google Scholar

[33]

R. Mañé, An ergodic closing lemma,, Ann. of Math., 116 (1982), 503.  doi: 10.2307/2007021.  Google Scholar

[34]

M. Pollicott, "Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds,", Cambridge Univ. Press, (1993).   Google Scholar

[35]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Ann. of Math. 151 (2000), 151 (2000), 961.   Google Scholar

[36]

M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents,, Invent. Math., 139 (2000), 495.  doi: 10.1007/s002229900035.  Google Scholar

[37]

P. Walters, "An Introduction to Ergodic Theory,", Springer, (1982).   Google Scholar

[38]

Z. Wang and W. Sun, Lyapunov exponents of hyperbolic measures andhyperbolic periodic orbits,, Trans. Amer. Math., 362 (2010), 4267.  doi: 10.1090/S0002-9947-10-04947-0.  Google Scholar

[39]

L. Wen, Homoclinic tangengcies and dominated splittings,, Nonlinearity, 15 (2002), 1445.  doi: 10.1088/0951-7715/15/5/306.  Google Scholar

[40]

L. Wen, A uniform $C^1$ connecting lemma,, Discrete and Continuous Dynamical Systems, 8 (2002), 257.   Google Scholar

[41]

L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensionalcycles,, Bull. Braz. Math. Soc., (2004), 419.  doi: 10.1007/s00574-004-0023-x.  Google Scholar

[42]

L. Wen and Z. Xia, $C^1$ connecting lemmas,, Trans. Amer. Math., 352 (2000), 5213.  doi: 10.1090/S0002-9947-00-02553-8.  Google Scholar

[43]

J. Yang, "$C^1$ Dynamics Far From Tangencies,", Ph.D thesis, (2008).   Google Scholar

[44]

J. Yang, Ergodic measures far away from tangencies,, preprint, (2009).   Google Scholar

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