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

Dense area-preserving homeomorphisms have zero Lyapunov exponents

Mário Bessa 1, and  César M. Silva 2,

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.
Keywords: Area-preserving homeomorphisms, topological dynamics., Lyapunov exponents.
Mathematics Subject Classification: Primary: 54H20, 28D05 Secondary: 37B9.
Citation: Mário Bessa, César M. Silva. Dense area-preserving homeomorphisms have zero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1016/S0166-8641(98)00041-8. Google Scholar

[2]

F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity of $C^1$ generic diffeomorphisms,, to appear in Israel Journal of Mathematics., (). Google Scholar

[3]

S. Alpern and V. Prasad, "Typical Dynamics of Volume Preserving Homeomorphisms,", Cambridge Tracts in Mathematics, 139 (2000). Google Scholar

[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. Google Scholar

[5]

E. Akin, M. Hurley and J. Kennedy, Dynamics of topologically generic homeomorphisms,, Mem. Amer. Math. Soc., 164 (2003). Google Scholar

[6]

A. Arbieto and J. Bochi, $L^p$-generic cocycles have one-point Lyapunov spectrum,, Stochastics and Dynamics, 3 (2003), 73. doi: 10.1142/S0219493703000619. Google Scholar

[7]

A. Arbieto and C. Matheus, A pasting lemma and some applications for conservative systems,, Ergodic Theory Dynam. Systems, 27 (2007), 1399. Google Scholar

[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. Google Scholar

[9]

V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Graduate Texts in Mathematics, 60 (1978). Google Scholar

[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. Google Scholar

[11]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacy, On Devaney's definition of chaos,, Amer. Math. Monthly, 99 (1992), 332. doi: 10.2307/2324899. Google Scholar

[12]

L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems,, Ergod. Th. & Dynam. Sys., 16 (1996), 871. Google Scholar

[13]

L. Barreira and Y. Pesin, "Lectures on Lyapunov Exponents and Smooth Ergodic Theory,", Proc. Sympos. Pure Math., 69 (1999), 3. Google Scholar

[14]

L. Barreira and C. Silva, Lyapunov exponents for continuous transformations and dimension theory,, Discrete Contin. Dyn. Syst., 13 (2005), 469. doi: 10.3934/dcds.2005.13.469. Google Scholar

[15]

J. Bochi, Genericity of zero Lyapunov exponents,, Ergod. Th. & Dynam. Sys., 22 (2002), 1667. Google Scholar

[16]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps,, Ann. of Math. (2), 161 (2005), 1423. Google Scholar

[17]

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

[18]

D. Bylov, R. Vinograd, D. Grobman and V. Nemyckiĭ, "Theory of Lyapunov Exponents and its Application to Problems of Stability,'', in Russian, (1966). Google Scholar

[19]

F. Daalderop and R. Fokkink, Chaotic homeomorphisms are generic,, Topology Appl., 102 (2000), 297. doi: 10.1016/S0166-8641(98)00155-2. Google Scholar

[20]

M. K. Fort, Category theorems,, Fund. Math., 42 (1955), 276. Google Scholar

[21]

M. Hurley, On proofs of the $C^0$ general density theorem,, Proc. Amer. Math. Soc., 124 (1996), 1305. doi: 10.1090/S0002-9939-96-03184-X. Google Scholar

[22]

A. Katok, Bernoulli diffeomorphisms on surfaces,, Ann. of Math. (2), 110 (1979), 529. doi: 10.2307/1971237. Google Scholar

[23]

P. Kościelniak, On genericity of chaos,, Topology Appl., 154 (2007), 1951. doi: 10.1016/j.topol.2007.01.014. Google Scholar

[24]

K. Kuratowski, "Topology," Vol. 1,, Academic Press, (1966). Google Scholar

[25]

Y. Kifer, Characteristic exponents of dynamical systems in metric spaces,, Ergodic Theory Dynam. Systems, 3 (1983), 119. doi: 10.1017/S0143385700001838. Google Scholar

[26]

R. Mañé, Oseledec's theorem from the generic viewpoint,, in, (1983), 1269. Google Scholar

[27]

S. Müller, Approximation of volume-preserving homeomorphisms by volume-preserving diffeomorphisms,, preprint, (2009). Google Scholar

[28]

J. Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms,, Ann. of Math. (2), 72 (1960), 521. doi: 10.2307/1970228. Google Scholar

[29]

Yong-Geun Oh, $C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphisms,, preprint, (2006). Google Scholar

[30]

V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197. Google Scholar

[31]

J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Ann. of Math. (2), 42 (1941), 874. doi: 10.2307/1968772. Google Scholar

[32]

J. Palis, C. Pugh, M. Shub and D. Sullivan, Genericity theorems in topological dynamics,, in, 468 (1974), 241. Google Scholar

[33]

M. Pollicott, "Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds,", London Mathematical Society Lecture Notes Series, 180 (1993). Google Scholar

[34]

C. Robinson, Generic properties of conservative systems,, Am. J. Math., 92 (1970), 562. doi: 10.2307/2373361. Google Scholar

[35]

J.-C. Sikorav, Approximation of a volume-preserving homeomorphism by a volume-preserving diffeomorphism,, symplexe, (2007). Google Scholar

[36]

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents,, Ann. of Math. (2), 167 (2008), 643. doi: 10.4007/annals.2008.167.643. Google Scholar

[37]

J.-C. Yoccoz, Travaux de Herman sur les Tores invariants,, Séminaire Bourbaki, 206 (1992), 311. Google Scholar

[38]

E. Zehnder, Note on smoothing symplectic and volume-preserving diffeomorphisms,, in, 597 (1976), 828. Google Scholar

show all references

References:
[1]

J. Aarts and F. Daalderop, Chaotic homeomorphisms on manifolds,, Topology Appl., 96 (1999), 93. doi: 10.1016/S0166-8641(98)00041-8. Google Scholar

[2]

F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity of $C^1$ generic diffeomorphisms,, to appear in Israel Journal of Mathematics., (). Google Scholar

[3]

S. Alpern and V. Prasad, "Typical Dynamics of Volume Preserving Homeomorphisms,", Cambridge Tracts in Mathematics, 139 (2000). Google Scholar

[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. Google Scholar

[5]

E. Akin, M. Hurley and J. Kennedy, Dynamics of topologically generic homeomorphisms,, Mem. Amer. Math. Soc., 164 (2003). Google Scholar

[6]

A. Arbieto and J. Bochi, $L^p$-generic cocycles have one-point Lyapunov spectrum,, Stochastics and Dynamics, 3 (2003), 73. doi: 10.1142/S0219493703000619. Google Scholar

[7]

A. Arbieto and C. Matheus, A pasting lemma and some applications for conservative systems,, Ergodic Theory Dynam. Systems, 27 (2007), 1399. Google Scholar

[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. Google Scholar

[9]

V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Graduate Texts in Mathematics, 60 (1978). Google Scholar

[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. Google Scholar

[11]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacy, On Devaney's definition of chaos,, Amer. Math. Monthly, 99 (1992), 332. doi: 10.2307/2324899. Google Scholar

[12]

L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems,, Ergod. Th. & Dynam. Sys., 16 (1996), 871. Google Scholar

[13]

L. Barreira and Y. Pesin, "Lectures on Lyapunov Exponents and Smooth Ergodic Theory,", Proc. Sympos. Pure Math., 69 (1999), 3. Google Scholar

[14]

L. Barreira and C. Silva, Lyapunov exponents for continuous transformations and dimension theory,, Discrete Contin. Dyn. Syst., 13 (2005), 469. doi: 10.3934/dcds.2005.13.469. Google Scholar

[15]

J. Bochi, Genericity of zero Lyapunov exponents,, Ergod. Th. & Dynam. Sys., 22 (2002), 1667. Google Scholar

[16]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps,, Ann. of Math. (2), 161 (2005), 1423. Google Scholar

[17]

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

[18]

D. Bylov, R. Vinograd, D. Grobman and V. Nemyckiĭ, "Theory of Lyapunov Exponents and its Application to Problems of Stability,'', in Russian, (1966). Google Scholar

[19]

F. Daalderop and R. Fokkink, Chaotic homeomorphisms are generic,, Topology Appl., 102 (2000), 297. doi: 10.1016/S0166-8641(98)00155-2. Google Scholar

[20]

M. K. Fort, Category theorems,, Fund. Math., 42 (1955), 276. Google Scholar

[21]

M. Hurley, On proofs of the $C^0$ general density theorem,, Proc. Amer. Math. Soc., 124 (1996), 1305. doi: 10.1090/S0002-9939-96-03184-X. Google Scholar

[22]

A. Katok, Bernoulli diffeomorphisms on surfaces,, Ann. of Math. (2), 110 (1979), 529. doi: 10.2307/1971237. Google Scholar

[23]

P. Kościelniak, On genericity of chaos,, Topology Appl., 154 (2007), 1951. doi: 10.1016/j.topol.2007.01.014. Google Scholar

[24]

K. Kuratowski, "Topology," Vol. 1,, Academic Press, (1966). Google Scholar

[25]

Y. Kifer, Characteristic exponents of dynamical systems in metric spaces,, Ergodic Theory Dynam. Systems, 3 (1983), 119. doi: 10.1017/S0143385700001838. Google Scholar

[26]

R. Mañé, Oseledec's theorem from the generic viewpoint,, in, (1983), 1269. Google Scholar

[27]

S. Müller, Approximation of volume-preserving homeomorphisms by volume-preserving diffeomorphisms,, preprint, (2009). Google Scholar

[28]

J. Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms,, Ann. of Math. (2), 72 (1960), 521. doi: 10.2307/1970228. Google Scholar

[29]

Yong-Geun Oh, $C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphisms,, preprint, (2006). Google Scholar

[30]

V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197. Google Scholar

[31]

J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Ann. of Math. (2), 42 (1941), 874. doi: 10.2307/1968772. Google Scholar

[32]

J. Palis, C. Pugh, M. Shub and D. Sullivan, Genericity theorems in topological dynamics,, in, 468 (1974), 241. Google Scholar

[33]

M. Pollicott, "Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds,", London Mathematical Society Lecture Notes Series, 180 (1993). Google Scholar

[34]

C. Robinson, Generic properties of conservative systems,, Am. J. Math., 92 (1970), 562. doi: 10.2307/2373361. Google Scholar

[35]

J.-C. Sikorav, Approximation of a volume-preserving homeomorphism by a volume-preserving diffeomorphism,, symplexe, (2007). Google Scholar

[36]

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents,, Ann. of Math. (2), 167 (2008), 643. doi: 10.4007/annals.2008.167.643. Google Scholar

[37]

J.-C. Yoccoz, Travaux de Herman sur les Tores invariants,, Séminaire Bourbaki, 206 (1992), 311. Google Scholar

[38]

E. Zehnder, Note on smoothing symplectic and volume-preserving diffeomorphisms,, in, 597 (1976), 828. Google Scholar

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