# American Institute of Mathematical Sciences

April  2012, 32(4): 1231-1244. doi: 10.3934/dcds.2012.32.1231

## Dense area-preserving homeomorphisms have zero Lyapunov exponents

 1 Departamento de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal 2 Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama, 6201-001 Covilhã, Portugal

Received  October 2010 Revised  July 2011 Published  October 2011

We give a new definition (different from the one in [14]) for a Lyapunov exponent (called new Lyapunov exponent) associated to a continuous map. Our first result states that these new exponents coincide with the usual Lyapunov exponents if the map is differentiable. Then, we apply this concept to prove that there exists a $C^0$-dense subset of the set of the area-preserving homeomorphisms defined in a compact, connected and boundaryless surface such that any element inside this residual subset has zero new Lyapunov exponents for Lebesgue almost every point. Finally, we prove that the function that associates an area-preserving homeomorphism, equipped with the $C^0$-topology, to the integral (with respect to area) of its top new Lyapunov exponent over the whole surface cannot be upper-semicontinuous.
Citation: Mário Bessa, César M. Silva. Dense area-preserving homeomorphisms have zero Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1231-1244. doi: 10.3934/dcds.2012.32.1231
##### References:
 [1] J. Aarts and F. Daalderop, Chaotic homeomorphisms on manifolds, Topology Appl., 96 (1999), 93-96. doi: 10.1016/S0166-8641(98)00041-8. [2] F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity of $C^1$ generic diffeomorphisms, to appear in Israel Journal of Mathematics. [3] S. Alpern and V. Prasad, "Typical Dynamics of Volume Preserving Homeomorphisms," Cambridge Tracts in Mathematics, 139, Cambridge University Press, Cambridge, 2000. [4] S. Alpern and V. Prasad, Properties generic for Lebesgue space automorphisms are generic for measure-preserving manifold homeomorphisms, Ergod. Th. & Dynam. Sys., 22 (2002), 1587-1620. [5] E. Akin, M. Hurley and J. Kennedy, Dynamics of topologically generic homeomorphisms, Mem. Amer. Math. Soc., 164 (2003), viii+130 pp. [6] A. Arbieto and J. Bochi, $L^p$-generic cocycles have one-point Lyapunov spectrum, Stochastics and Dynamics, 3 (2003), 73-81. doi: 10.1142/S0219493703000619. [7] A. Arbieto and C. Matheus, A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems, 27 (2007), 1399-1417. [8] 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. [9] V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer Verlag, New York-Heidelberg, 1978. [10] S. Baldwin and E. Slaminka, A stable/unstable "manifold" theorem for area preserving homeomorphisms of two manifolds, Proc. Amer. Math. Soc., 109 (1990), 823-828. [11] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacy, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334. doi: 10.2307/2324899. [12] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., 16 (1996), 871-927. [13] L. Barreira and Y. Pesin, "Lectures on Lyapunov Exponents and Smooth Ergodic Theory," Proc. Sympos. Pure Math., 69, Smooth Ergotic Theory and its Applications (Seattle, WA, 1999), 3-106, Amer. Math. Soc., Providence, RI, 2001. [14] L. Barreira and C. Silva, Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dyn. Syst., 13 (2005), 469-490. doi: 10.3934/dcds.2005.13.469. [15] J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Th. & Dynam. Sys., 22 (2002), 1667-1696. [16] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. of Math. (2), 161 (2005), 1423-1485. [17] C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective," Encycl. of Math. Sc., 102, Math. Phys., III, Springer-Verlag, Berlin, 2005. [18] D. Bylov, R. Vinograd, D. Grobman and V. Nemyckiĭ , "Theory of Lyapunov Exponents and its Application to Problems of Stability,'' in Russian, Izdat. "Nauka,'' Moscow, 1966. [19] F. Daalderop and R. Fokkink, Chaotic homeomorphisms are generic, Topology Appl., 102 (2000), 297-302. doi: 10.1016/S0166-8641(98)00155-2. [20] M. K. Fort, Category theorems, Fund. Math., 42 (1955), 276-288. [21] M. Hurley, On proofs of the $C^0$ general density theorem, Proc. Amer. Math. Soc., 124 (1996), 1305-1309. doi: 10.1090/S0002-9939-96-03184-X. [22] A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2), 110 (1979), 529-547. doi: 10.2307/1971237. [23] P. Kościelniak, On genericity of chaos, Topology Appl., 154 (2007), 1951-1955. doi: 10.1016/j.topol.2007.01.014. [24] K. Kuratowski, "Topology," Vol. 1, Academic Press, New York-London, Państwowe Wydawnictwo Naukowe, Warsaw, 1966. [25] Y. Kifer, Characteristic exponents of dynamical systems in metric spaces, Ergodic Theory Dynam. Systems, 3 (1983), 119-127. doi: 10.1017/S0143385700001838. [26] R. Mañé, Oseledec's theorem from the generic viewpoint, in "Proceedings of the International Congress of Mathematicians," Vol. 1, 2 (Warsaw, 1983), 1269-1276, PWN, Warsaw, 1984. [27] S. Müller, Approximation of volume-preserving homeomorphisms by volume-preserving diffeomorphisms, preprint, arXiv:0901.1002, 2009. [28] J. Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. (2), 72 (1960), 521-554. doi: 10.2307/1970228. [29] Yong-Geun Oh, $C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphisms, preprint, arXiv:math/0601183, 2006. [30] V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231. [31] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), 42 (1941), 874-920. doi: 10.2307/1968772. [32] J. Palis, C. Pugh, M. Shub and D. Sullivan, Genericity theorems in topological dynamics, in "Dynamical systems-Warwick 1974" (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974, presented to E. C. Zeeman on his fiftieth birthday), 241-250, Lecture Notes in Math., 468, Springer, Berlin, 1975. [33] M. Pollicott, "Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds," London Mathematical Society Lecture Notes Series, 180, Cambridge University Press, Cambridge, 1993. [34] C. Robinson, Generic properties of conservative systems, Am. J. Math., 92 (1970), 562-603. doi: 10.2307/2373361. [35] J.-C. Sikorav, Approximation of a volume-preserving homeomorphism by a volume-preserving diffeomorphism, symplexe, September 2007. Avaiable from: http://www.umpa.ens-lyon.fr/. [36] M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. of Math. (2), 167 (2008), 643-680. doi: 10.4007/annals.2008.167.643. [37] J.-C. Yoccoz, Travaux de Herman sur les Tores invariants, Séminaire Bourbaki, Vol. 1991/92, Astérisque, 206 (1992), 311-344. [38] E. Zehnder, Note on smoothing symplectic and volume-preserving diffeomorphisms, in "Geometry and Topology" (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), 828-854, Lecture Notes in Math., 597, Springer, Berlin, 1977.

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##### References:
 [1] J. Aarts and F. Daalderop, Chaotic homeomorphisms on manifolds, Topology Appl., 96 (1999), 93-96. doi: 10.1016/S0166-8641(98)00041-8. [2] F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity of $C^1$ generic diffeomorphisms, to appear in Israel Journal of Mathematics. [3] S. Alpern and V. Prasad, "Typical Dynamics of Volume Preserving Homeomorphisms," Cambridge Tracts in Mathematics, 139, Cambridge University Press, Cambridge, 2000. [4] S. Alpern and V. Prasad, Properties generic for Lebesgue space automorphisms are generic for measure-preserving manifold homeomorphisms, Ergod. Th. & Dynam. Sys., 22 (2002), 1587-1620. [5] E. Akin, M. Hurley and J. Kennedy, Dynamics of topologically generic homeomorphisms, Mem. Amer. Math. Soc., 164 (2003), viii+130 pp. [6] A. Arbieto and J. Bochi, $L^p$-generic cocycles have one-point Lyapunov spectrum, Stochastics and Dynamics, 3 (2003), 73-81. doi: 10.1142/S0219493703000619. [7] A. Arbieto and C. Matheus, A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems, 27 (2007), 1399-1417. [8] 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. [9] V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer Verlag, New York-Heidelberg, 1978. [10] S. Baldwin and E. Slaminka, A stable/unstable "manifold" theorem for area preserving homeomorphisms of two manifolds, Proc. Amer. Math. Soc., 109 (1990), 823-828. [11] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacy, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334. doi: 10.2307/2324899. [12] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., 16 (1996), 871-927. [13] L. Barreira and Y. Pesin, "Lectures on Lyapunov Exponents and Smooth Ergodic Theory," Proc. Sympos. Pure Math., 69, Smooth Ergotic Theory and its Applications (Seattle, WA, 1999), 3-106, Amer. Math. Soc., Providence, RI, 2001. [14] L. Barreira and C. Silva, Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dyn. Syst., 13 (2005), 469-490. doi: 10.3934/dcds.2005.13.469. [15] J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Th. & Dynam. Sys., 22 (2002), 1667-1696. [16] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. of Math. (2), 161 (2005), 1423-1485. [17] C. Bonatti, L. J. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective," Encycl. of Math. Sc., 102, Math. Phys., III, Springer-Verlag, Berlin, 2005. [18] D. Bylov, R. Vinograd, D. Grobman and V. Nemyckiĭ , "Theory of Lyapunov Exponents and its Application to Problems of Stability,'' in Russian, Izdat. "Nauka,'' Moscow, 1966. [19] F. Daalderop and R. Fokkink, Chaotic homeomorphisms are generic, Topology Appl., 102 (2000), 297-302. doi: 10.1016/S0166-8641(98)00155-2. [20] M. K. Fort, Category theorems, Fund. Math., 42 (1955), 276-288. [21] M. Hurley, On proofs of the $C^0$ general density theorem, Proc. Amer. Math. Soc., 124 (1996), 1305-1309. doi: 10.1090/S0002-9939-96-03184-X. [22] A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2), 110 (1979), 529-547. doi: 10.2307/1971237. [23] P. Kościelniak, On genericity of chaos, Topology Appl., 154 (2007), 1951-1955. doi: 10.1016/j.topol.2007.01.014. [24] K. Kuratowski, "Topology," Vol. 1, Academic Press, New York-London, Państwowe Wydawnictwo Naukowe, Warsaw, 1966. [25] Y. Kifer, Characteristic exponents of dynamical systems in metric spaces, Ergodic Theory Dynam. Systems, 3 (1983), 119-127. doi: 10.1017/S0143385700001838. [26] R. Mañé, Oseledec's theorem from the generic viewpoint, in "Proceedings of the International Congress of Mathematicians," Vol. 1, 2 (Warsaw, 1983), 1269-1276, PWN, Warsaw, 1984. [27] S. Müller, Approximation of volume-preserving homeomorphisms by volume-preserving diffeomorphisms, preprint, arXiv:0901.1002, 2009. [28] J. Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. (2), 72 (1960), 521-554. doi: 10.2307/1970228. [29] Yong-Geun Oh, $C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphisms, preprint, arXiv:math/0601183, 2006. [30] V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231. [31] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), 42 (1941), 874-920. doi: 10.2307/1968772. [32] J. Palis, C. Pugh, M. Shub and D. Sullivan, Genericity theorems in topological dynamics, in "Dynamical systems-Warwick 1974" (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974, presented to E. C. Zeeman on his fiftieth birthday), 241-250, Lecture Notes in Math., 468, Springer, Berlin, 1975. [33] M. Pollicott, "Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds," London Mathematical Society Lecture Notes Series, 180, Cambridge University Press, Cambridge, 1993. [34] C. Robinson, Generic properties of conservative systems, Am. J. Math., 92 (1970), 562-603. doi: 10.2307/2373361. [35] J.-C. Sikorav, Approximation of a volume-preserving homeomorphism by a volume-preserving diffeomorphism, symplexe, September 2007. Avaiable from: http://www.umpa.ens-lyon.fr/. [36] M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. of Math. (2), 167 (2008), 643-680. doi: 10.4007/annals.2008.167.643. [37] J.-C. Yoccoz, Travaux de Herman sur les Tores invariants, Séminaire Bourbaki, Vol. 1991/92, Astérisque, 206 (1992), 311-344. [38] E. Zehnder, Note on smoothing symplectic and volume-preserving diffeomorphisms, in "Geometry and Topology" (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), 828-854, Lecture Notes in Math., 597, Springer, Berlin, 1977.
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