Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-31T09:15:12.911Z Has data issue: false hasContentIssue false
  • English
  • Français

A geometric path from zero Lyapunov exponents to rotation cocycles

Published online by Cambridge University Press:  20 August 2013

JAIRO BOCHI  and
ANDRÉS NAVAS
JAIRO BOCHI
Affiliation:
PUC–Rio, Rua Marquês de S. Vicente, 225 Rio de Janeiro, Brazil email jairo@mat.puc-rio.br
ANDRÉS NAVAS
Affiliation:
Universidad de Santiago, Alameda 3363, Estación Central, Santiago, Chile email andres.navas@usach.cl
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider cocycles of isometries on spaces of non-positive curvature $H$. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there are almost invariant sections, that is, sections that move arbitrarily little under the cocycle dynamics. If, in addition, $H$ is a symmetric space, then we show that almost invariant sections can be made invariant by perturbing the cocycle.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 2 , April 2015 , pp. 374 - 402
Copyright
© Cambridge University Press, 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aleksandrov, A. D., Berestovskiĭ, V. N. and Nikolaev, I. G.. Generalized Riemannian spaces. Russian Math. Surveys 41 (3) (1986), 1–54.Google Scholar
Arnaudon, M. and Li, X.-M.. Barycenters of measures transported by stochastic flows. Ann. Probab. 33 (4) (2005), 1509–1543.Google Scholar
Avila, A. and Bochi, J.. A uniform dichotomy for generic cocycles over a minimal base. Bull. Soc. Math. France 135 (3) (2007), 407–417.CrossRefGoogle Scholar
Avila, A., Bochi, J. and Damanik, D.. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J. 146 (2) (2009), 253–280.Google Scholar
Avila, A., Bochi, J. and Damanik, D.. Opening gaps in the spectrum of strictly ergodic Schrödinger operators. J. Eur. Math. Soc. 14 (1) (2012), 61–106.Google Scholar
Avila, A. and Kocsard, A.. Cohomological equations and invariant distributions for minimal circle diffeomorphisms. Duke Math. J. 158 (2) (2011), 501–536.Google Scholar
Ballmann, W., Gromov, M. and Schroeder, V.. Manifolds of Nonpositive Curvature (Progress in Mathematics, 61). Birkhäuser, Boston, 1985.CrossRefGoogle Scholar
Biliotti, L.. The exponential map of a weak Riemannian Hilbert manifold. Illinois J. Math. 48 (4) (2004), 1191–1206.Google Scholar
Bochi, J. and Navas, A.. Almost reduction and perturbation of matrix cocycles. Preprint arXiv:1301.5464.Google Scholar
Bonatti, C. and Guelman, N.. Smooth conjugacy classes of circle diffeomorphisms with irrational rotation number. Preprint arXiv:1207.2508.Google Scholar
Breuillard, E. and Green, B.. Approximate groups. I: The torsion-free nilpotent case. J. Inst. Math. Jussieu 10 (1) (2011), 37–57.Google Scholar
Bridson, M. R. and Haefliger, A.. Metric Spaces of Non-positive Curvature (Grundlehren der mathematischen Wissenschaften, 319). Springer, Berlin, 1999.CrossRefGoogle Scholar
Burago, D., Burago, Yu. and Ivanov, S.. A Course in Metric Geometry (Graduate Studies in Mathematics, 33). American Mathematical Society, Providence, RI, 2001.CrossRefGoogle Scholar
Buser, P. and Karcher, H.. Gromov’s Almost Flat Manifolds (Astérisque, 81). Société Mathématique de France, Paris, 1981.Google Scholar
Cartan, E.. Leçons sur la géométrie des espaces de Riemann. Gauthier-Villars, Paris, 1928.Google Scholar
Cheeger, J. and Ebin, D. G.. Comparison Theorems in Riemannian Geometry. AMS Chelsea Publishing, Providence, RI, 2008, revised reprint of the 1975 original.Google Scholar
Corach, G., Porta, H. and Recht, L.. A geometric interpretation of Segal’s inequality $\Vert {e}^{X+ Y} \Vert \leq \Vert {e}^{X/ 2} {e}^{Y} {e}^{X/ 2} \Vert $. Proc. Amer. Math. Soc. 115 (1) (1992), 229–231.Google Scholar
Coronel, D., Navas, A. and Ponce, M.. On the dynamics of nonreducible cylindrical vortices. J. Lond. Math. Soc. 85 (3) (2012), 789–808.Google Scholar
de Cornulier, Y., Tessera, R. and Valette, A.. Isometric group actions on Hilbert spaces: growth of cocycles. Geom. Funct. Anal. 17 (2007), 770–792.Google Scholar
Es-Sahib, A. and Heinich, H.. Barycentre canonique pour un espace métrique à courbure négative. Séminaire de Probabilités XXXIII (Lecture Notes in Mathematics, 1709). Springer, Berlin, 1999, pp. 355–370.Google Scholar
Helgason, S.. Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics, 34). American Mathematical Society, Providence, RI, 2001, corrected reprint of the 1978 original.Google Scholar
Jenkinson, O.. Ergodic optimization. Discrete Contin. Dyn. Sys. 15 (1) (2006), 197–224.Google Scholar
Jost, J.. Nonpositive Curvature: Geometric and Analytic Aspects (Lectures in Mathematics—ETH Zürich). Birkhäuser, Basel, 1997.Google Scholar
Karcher, H.. Riemannian comparison constructions. Global Differential Geometry (MAA Studies in Mathematics, 27). Mathematical Association of America, Washington, DC, 1989, pp. 170–222.Google Scholar
Karlsson, A. and Margulis, G. A.. A multiplicative ergodic theorem and nonpositively curved spaces. Comm. Math. Phys. 208 (1) (1999), 107–123.Google Scholar
Katok, A.. Cocycles, cohomology and combinatorial constructions in ergodic theory. Smooth Ergodic Theory and Its Applications (Proceedings of Symposia in Pure Mathematics, 69). Eds. Katok, A., de la Llave, R., Pesin, Ya. and Weiss, H.. American Mathematical Society, Providence, RI, 2001, pp. 107–173; in collaboration with E. A. Robinson Jr.Google Scholar
Kobayashi, S. and Nomizu, K.. Foundations of Differential Geometry. Vol. II (Interscience Tracts in Pure and Applied Mathematics, 15). John Wiley & Sons, New York–London–Sydney, 1969.Google Scholar
Lang, S.. Fundamentals of Differential Geometry (Graduate Texts in Mathematics, 191). Springer, New York, 1999.Google Scholar
Larotonda, G.. Nonpositive curvature: a geometrical approach to Hilbert–Schmidt operators. Differ. Geom. Appl. 25 (2007), 679–700.CrossRefGoogle Scholar
Lawson, J. and Lim, Y.. Symmetric spaces with convex metrics. Forum Math. 19 (4) (2007), 571–602.CrossRefGoogle Scholar
Moulin Ollagnier, J. and Pinchon, D.. Mesures quasi invariantes à dérivée continue. C. R. Acad. Sci. Paris Sér. A–B 282 (23) (1976), Aii, A1371–A1373.Google Scholar
Moulin Ollagnier, J. and Pinchon, D.. Systèmes dynamiques topologiques I. Étude des limites de cobords. Bull. Soc. Math. France 105 (1977), 405–414.Google Scholar
Navas, A.. An ${L}^{1} $ ergodic theorem with values in a nonpositively curved space via a canonical barycenter map. Erg. Th. & Dynam. Sys. 33 (2) (2013), 609–623.CrossRefGoogle Scholar
Navas, A.. Sur les rapprochements par conjugaison en dimension 1 et classe ${C}^{1} $. Preprint arXiv:1208.4815.Google Scholar
Navas, A. and Triestino, M.. On the invariant distributions of ${C}^{2} $ circle diffeomorphisms of irrational rotation number. Math. Z. 274 (2013), 315–321.CrossRefGoogle Scholar
Papadopoulos, A.. Metric Spaces, Convexity and Nonpositive Curvature (IRMA Lectures in Mathematics and Theoretical Physics, 6). EMS, Zürich, 2005.Google Scholar
Schreiber, S. J.. On growth rates of subadditive functions for semiflows. J. Differential Equations 148 (1998), 334–350.Google Scholar
Schwartzman, S.. Asymptotic cycles. Ann. Math. 66 (1957), 270–284.CrossRefGoogle Scholar
Sturm, K.-T.. Probability measures on metric spaces of nonpositive curvature. Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Contemporary Mathematics, 338). Eds. Auscher, Coulhon and Grigor’yan, . American Mathematical Society, Providence, RI, 2003, pp. 357–390.Google Scholar
Sturman, R. and Stark, J.. Semi-uniform ergodic theorems and applications to forced systems. Nonlinearity 13 (2000), 113–143.Google Scholar
Villani, C.. Optimal Transport, Old and New (Grundlehren der Mathematischen Wissenschaften, 338). Springer, Berlin, 2009.Google Scholar