Singular hyperbolicity and sectional Lyapunov exponents of various orders

Salgado, Luciana

doi:10.1090/proc/14254

Singular hyperbolicity and sectional Lyapunov exponents of various orders
HTML articles powered by AMS MathViewer

by Luciana Salgado PDF

Proc. Amer. Math. Soc. 147 (2019), 735-749 Request permission

Abstract:

There are notions of singular hyperbolicity and sectional Lyapunov exponents of orders beyond the classical ones, namely, other dimensions besides the dimension 2 and the full dimension of the central subbundle of the singular hyperbolic set. We obtain a characterization of dominated splittings, partial and singular hyperbolicity in this broad sense, by using Lyapunov exponents and the notion of infinitesimal Lyapunov functions. Furthermore, we give alternative requirements to obtain singular hyperbolicity. As an application we obtain some results related to singular hyperbolic sets for flows.

References

Vitor Araujo and Luciana Salgado, Infinitesimal Lyapunov functions for singular flows, Math. Z. 275 (2013), no. 3-4, 863–897. MR 3127040, DOI 10.1007/s00209-013-1163-8
Vitor Araujo and Luciana Salgado, Dominated splitting for exterior powers and singular hyperbolicity, J. Differential Equations 259 (2015), no. 8, 3874–3893. MR 3369265, DOI 10.1016/j.jde.2015.05.006
Vitor Araujo, Alexander Arbieto, and Luciana Salgado, Dominated splittings for flows with singularities, Nonlinearity 26 (2013), no. 8, 2391–2407. MR 3084717, DOI 10.1088/0951-7715/26/8/2391
Alexander Arbieto and Luciana Salgado, On critical orbits and sectional hyperbolicity of the nonwandering set for flows, J. Differential Equations 250 (2011), no. 6, 2927–2939. MR 2771270, DOI 10.1016/j.jde.2011.01.023
Vítor Araújo and Maria José Pacifico, Three-dimensional flows, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 53, Springer, Heidelberg, 2010. With a foreword by Marcelo Viana. MR 2662317, DOI 10.1007/978-3-642-11414-4
Alexander Arbieto, Sectional Lyapunov exponents, Proc. Amer. Math. Soc. 138 (2010), no. 9, 3171–3178. MR 2653942, DOI 10.1090/S0002-9939-10-10410-9
Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
Christian Bonatti, Lorenzo J. Díaz, and Marcelo Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective; Mathematical Physics, III. MR 2105774
Nikolaz Gourmelon, Adapted metrics for dominated splittings, Ergodic Theory Dynam. Systems 27 (2007), no. 6, 1839–1849. MR 2371598, DOI 10.1017/S0143385707000272
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
Anatole Katok, Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems, Ergodic Theory Dynam. Systems 14 (1994), no. 4, 757–785. With the collaboration of Keith Burns. MR 1304141, DOI 10.1017/S0143385700008142
J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B 30 (1968), 499–510. MR 254907
Jorge Lewowicz, Lyapunov functions and topological stability, J. Differential Equations 38 (1980), no. 2, 192–209. MR 597800, DOI 10.1016/0022-0396(80)90004-2
A. I. Mal′cev, Foundations of linear algebra, W. H. Freeman and Co., San Francisco, Calif.-London, 1963. Translated from the Russian by Thomas Craig Brown; edited by J. B. Roberts. MR 0166200
Ricardo Mañé, An ergodic closing lemma, Ann. of Math. (2) 116 (1982), no. 3, 503–540. MR 678479, DOI 10.2307/2007021
R. Metzger and C. Morales, Sectional-hyperbolic systems, Ergodic Theory Dynam. Systems 28 (2008), no. 5, 1587–1597. MR 2449545, DOI 10.1017/S0143385707000995
C. A. Morales, M. J. Pacifico, and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3393–3401. MR 1610761, DOI 10.1090/S0002-9939-99-04936-9
C. A. Morales, M. J. Pacifico, and E. R. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math. (2) 160 (2004), no. 2, 375–432. MR 2123928, DOI 10.4007/annals.2004.160.375
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
V. P. Potapov, The multiplicative structure of $J$-contractive matrix functions, Trudy Moskov. Mat. Obšč. 4 (1955), 125–236 (Russian). MR 0076882
V. P. Potapov, Linear-fractional transformations of matrices, Studies in the theory of operators and their applications (Russian), “Naukova Dumka”, Kiev, 1979, pp. 75–97, 177 (Russian). MR 566141
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
Maciej Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergodic Theory Dynam. Systems 5 (1985), no. 1, 145–161. MR 782793, DOI 10.1017/S0143385700002807
Maciej P. Wojtkowski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature, Fund. Math. 163 (2000), no. 2, 177–191. MR 1752103, DOI 10.4064/fm-163-2-177-191
Maciej P. Wojtkowski, Monotonicity, $\scr J$-algebra of Potapov and Lyapunov exponents, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 499–521. MR 1858544, DOI 10.1090/pspum/069/1858544
Maciej P. Wojtkowski, A simple proof of polar decomposition in pseudo-Euclidean geometry, Fund. Math. 206 (2009), 299–306. MR 2576274, DOI 10.4064/fm206-0-18