# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 1-22. doi: 10.1017/S0143385706000538. [2] A. Arbieto and T. Catalan, Hyperbolicity in the volume preserving scenario,, preprint, (). [3] M. C. Arnaud, Cr$\acutee$ation de connexions en topologie $C^1$, (French) [Creating connections in the $C^1$-topology], Ergod. Th. Dynam. Sys., 21 (2001), 339-381. [4] L. Arnold, "Random Dynamical Systems," Springer-Verlag, New York, 1998. [5] L. Arnold and N. D. Cong, On the simplicity of the Lyapunov spectrum ofproducts of random matrices, Ergod. Th. Dynam. Sys., 17 (1997), 1005-1025. doi: 10.1017/S0143385797086355. [6] L. Arnold and N. D. Cong, Linear cocycles with simple Lyapunov spectrum are dense in $L^\infty$, Ergod. Th. Dynam. Sys., 19 (1999), 1389-1404. doi: 10.1017/S014338579915199X. [7] A. Ávila, On the regularization of conservative maps, Acta Mathematica, 205 (2010), 5-18. [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), 931-979. [9] A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents, Ergod. Th. Dynam. Sys., 22 (2003), 1655-1670. doi: 10.1017/S0143385702001773. [10] L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," A. M. S., 2002. [11] J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Th. Dynam. Sys., 22 (2002), 1667-1696. doi: 10.1017/S0143385702001165. [12] J. Bochi, B. R. Fayad and E. Pujals, A remark on conservative diffeomorphisms, C.R.Math. Acad. Sci. Paris, 342 (2006), 763-766 (English, with English and French summaries). [13] J. Bochi and M. Viana, The Lyapunov exponents of genericvolume-preserving and symplectic maps, Ann. of Math., 161 (2005), 1423-1458. doi: 10.4007/annals.2005.161.1423. [14] C. Bonatti and S. Crovisier, Récurrence et généricité, (French) [Recurrence and genericity], Invent. Math., 158 (2004), 33-104. [15] C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396. doi: 10.2307/2118647. [16] C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics, Journal of the Inst. of Math. Jussieu, 7 (2008), 469-525. [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), 355-418. doi: 10.4007/annals.2003.158.355. [18] C. Bonatti, L. J. Díaz, E.R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles, Astérisque, 286 (2003), 187-222. [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), 713-718. [20] C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity:A Global Geometric and Probabilistic Perspective," Encyclopaedia of MathematicalSciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005. [21] C. Bonatti and M. Viana, Lyapunov exponents with multiplicity 1 fordeterministic products of matrices, Ergod. Th. Dynam. Sys., 24 (2004), 1295-1330. doi: 10.1017/S0143385703000695. [22] K. Burns, D. Dolgopyat and Y. Pesin, Partial hyperbolicity, Lyapunovexponents and stable ergodicity, J. Stat. Phys., 109 (2002), 927-942. doi: 10.1023/A:1019779128351. [23] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic sysytems, Ann. of Math., 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451. [24] L. J. Díaz, Robust nonhyperbolic dynamics and heterodimensional cycles, Ergod. Th. Dynam. Sys., 15 (1995), 291-315. [25] D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense, Geometric Methods in Dynamics, II, Astérisque, 287 (2003), 33-60. [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), 81-137. [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), 599-629. doi: 10.1215/00127094-1444314. [28] F. R. Hertz, M. A. R. Hertz, A. Tahzibi and R. Ures, Creation of Blenders in the conservative setting, Nonlinearity (Bristol), 23 (2010), 211-223. [29] O. Knill, Positive Lyapunov indices determine absolutely continuousspectra of stationary random one-dimensional Schrodinger operators, Stochastic analysis, North Holland, (1984), 225-248. [30] C. Liang and G. Liu, Dominated splitting versus Small angles, Acta Math. Sinica, 24 (2008), 1163-1174. doi: 10.1007/s10114-007-6445-9. [31] C. Liang and G. Liu, Conditions for dominated splitting, Acta Math. Sinica, 25 (2009), 1389-1398. doi: 10.1007/s10114-009-6568-2. [32] C. Liang, G. Liu and W. Sun, Equivalent conditions fordominated splitting in volume-preserving diffeomorphisms, Acta Math. Sinica, 23 (2007), 1563-1576. doi: 10.1007/s10114-005-0889-6. [33] R. Mañé, An ergodic closing lemma, Ann. of Math., 116 (1982), 503-540. doi: 10.2307/2007021. [34] M. Pollicott, "Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds," Cambridge Univ. Press, 1993. [35] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. 151 (2000), 961-1023. [36] M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents, Invent. Math., 139 (2000), 495-508. doi: 10.1007/s002229900035. [37] P. Walters, "An Introduction to Ergodic Theory," Springer, 1982. [38] Z. Wang and W. Sun, Lyapunov exponents of hyperbolic measures andhyperbolic periodic orbits, Trans. Amer. Math., 362 (2010), 4267-4282. doi: 10.1090/S0002-9947-10-04947-0. [39] L. Wen, Homoclinic tangengcies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. doi: 10.1088/0951-7715/15/5/306. [40] L. Wen, A uniform $C^1$ connecting lemma, Discrete and Continuous Dynamical Systems, 8 (2002), 257-265. [41] L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensionalcycles, Bull. Braz. Math. Soc., \textbf {35} (2004), 419-452. doi: 10.1007/s00574-004-0023-x. [42] L. Wen and Z. Xia, $C^1$ connecting lemmas, Trans. Amer. Math., 352 (2000), 5213-5230. doi: 10.1090/S0002-9947-00-02553-8. [43] J. Yang, "$C^1$ Dynamics Far From Tangencies," Ph.D thesis, IMPA, 2008. Available from: http://www.preprint.impa.br. [44] J. Yang, Ergodic measures far away from tangencies, preprint, 2009. Available from: http://www.preprint.impa.br.

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##### 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), 1-22. doi: 10.1017/S0143385706000538. [2] A. Arbieto and T. Catalan, Hyperbolicity in the volume preserving scenario,, preprint, (). [3] M. C. Arnaud, Cr$\acutee$ation de connexions en topologie $C^1$, (French) [Creating connections in the $C^1$-topology], Ergod. Th. Dynam. Sys., 21 (2001), 339-381. [4] L. Arnold, "Random Dynamical Systems," Springer-Verlag, New York, 1998. [5] L. Arnold and N. D. Cong, On the simplicity of the Lyapunov spectrum ofproducts of random matrices, Ergod. Th. Dynam. Sys., 17 (1997), 1005-1025. doi: 10.1017/S0143385797086355. [6] L. Arnold and N. D. Cong, Linear cocycles with simple Lyapunov spectrum are dense in $L^\infty$, Ergod. Th. Dynam. Sys., 19 (1999), 1389-1404. doi: 10.1017/S014338579915199X. [7] A. Ávila, On the regularization of conservative maps, Acta Mathematica, 205 (2010), 5-18. [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), 931-979. [9] A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents, Ergod. Th. Dynam. Sys., 22 (2003), 1655-1670. doi: 10.1017/S0143385702001773. [10] L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," A. M. S., 2002. [11] J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Th. Dynam. Sys., 22 (2002), 1667-1696. doi: 10.1017/S0143385702001165. [12] J. Bochi, B. R. Fayad and E. Pujals, A remark on conservative diffeomorphisms, C.R.Math. Acad. Sci. Paris, 342 (2006), 763-766 (English, with English and French summaries). [13] J. Bochi and M. Viana, The Lyapunov exponents of genericvolume-preserving and symplectic maps, Ann. of Math., 161 (2005), 1423-1458. doi: 10.4007/annals.2005.161.1423. [14] C. Bonatti and S. Crovisier, Récurrence et généricité, (French) [Recurrence and genericity], Invent. Math., 158 (2004), 33-104. [15] C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396. doi: 10.2307/2118647. [16] C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics, Journal of the Inst. of Math. Jussieu, 7 (2008), 469-525. [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), 355-418. doi: 10.4007/annals.2003.158.355. [18] C. Bonatti, L. J. Díaz, E.R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles, Astérisque, 286 (2003), 187-222. [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), 713-718. [20] C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity:A Global Geometric and Probabilistic Perspective," Encyclopaedia of MathematicalSciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005. [21] C. Bonatti and M. Viana, Lyapunov exponents with multiplicity 1 fordeterministic products of matrices, Ergod. Th. Dynam. Sys., 24 (2004), 1295-1330. doi: 10.1017/S0143385703000695. [22] K. Burns, D. Dolgopyat and Y. Pesin, Partial hyperbolicity, Lyapunovexponents and stable ergodicity, J. Stat. Phys., 109 (2002), 927-942. doi: 10.1023/A:1019779128351. [23] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic sysytems, Ann. of Math., 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451. [24] L. J. Díaz, Robust nonhyperbolic dynamics and heterodimensional cycles, Ergod. Th. Dynam. Sys., 15 (1995), 291-315. [25] D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense, Geometric Methods in Dynamics, II, Astérisque, 287 (2003), 33-60. [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), 81-137. [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), 599-629. doi: 10.1215/00127094-1444314. [28] F. R. Hertz, M. A. R. Hertz, A. Tahzibi and R. Ures, Creation of Blenders in the conservative setting, Nonlinearity (Bristol), 23 (2010), 211-223. [29] O. Knill, Positive Lyapunov indices determine absolutely continuousspectra of stationary random one-dimensional Schrodinger operators, Stochastic analysis, North Holland, (1984), 225-248. [30] C. Liang and G. Liu, Dominated splitting versus Small angles, Acta Math. Sinica, 24 (2008), 1163-1174. doi: 10.1007/s10114-007-6445-9. [31] C. Liang and G. Liu, Conditions for dominated splitting, Acta Math. Sinica, 25 (2009), 1389-1398. doi: 10.1007/s10114-009-6568-2. [32] C. Liang, G. Liu and W. Sun, Equivalent conditions fordominated splitting in volume-preserving diffeomorphisms, Acta Math. Sinica, 23 (2007), 1563-1576. doi: 10.1007/s10114-005-0889-6. [33] R. Mañé, An ergodic closing lemma, Ann. of Math., 116 (1982), 503-540. doi: 10.2307/2007021. [34] M. Pollicott, "Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds," Cambridge Univ. Press, 1993. [35] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. 151 (2000), 961-1023. [36] M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents, Invent. Math., 139 (2000), 495-508. doi: 10.1007/s002229900035. [37] P. Walters, "An Introduction to Ergodic Theory," Springer, 1982. [38] Z. Wang and W. Sun, Lyapunov exponents of hyperbolic measures andhyperbolic periodic orbits, Trans. Amer. Math., 362 (2010), 4267-4282. doi: 10.1090/S0002-9947-10-04947-0. [39] L. Wen, Homoclinic tangengcies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. doi: 10.1088/0951-7715/15/5/306. [40] L. Wen, A uniform $C^1$ connecting lemma, Discrete and Continuous Dynamical Systems, 8 (2002), 257-265. [41] L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensionalcycles, Bull. Braz. Math. Soc., \textbf {35} (2004), 419-452. doi: 10.1007/s00574-004-0023-x. [42] L. Wen and Z. Xia, $C^1$ connecting lemmas, Trans. Amer. Math., 352 (2000), 5213-5230. doi: 10.1090/S0002-9947-00-02553-8. [43] J. Yang, "$C^1$ Dynamics Far From Tangencies," Ph.D thesis, IMPA, 2008. Available from: http://www.preprint.impa.br. [44] J. Yang, Ergodic measures far away from tangencies, preprint, 2009. Available from: http://www.preprint.impa.br.
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