Abstract
Given a vector fieldX on a Riemannian manifoldM of dimension at least 2 whose flow leaves a probability measureμ invariant, the multiplicative ergodic theorem tells us thatμ-a.s. every tangent vector possesses a Lyapunov exponent (exponential growth rate) that is equal to one of finitely many basic exponents corresponding toX andμ. We prove that, in the case of a simple Lyapunov spectrum, every tangent planeμ-a.s. possesses a rotation number that is equal to one of finitely many basic rotation numbers corresponding toX andμ. Rotation in a plane is measured as the time average of the infinitesimal changes of the angle between a frame moved by the linearized flow and the same frame parallel-transported by a (canonical) connection.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, L., and Kloeden, P. (1989). Lyapunov exponents and rotation number of two-dimensional systems with telegraphic noise. Report 189, Institute for Dynamical Systems, Bremen University, Bremen, to appear inSIAM J. Appl. Math.
Arnold, L., and Wihstutz, V. (eds.). (1986). Lyapunov exponents.Lect. Notes Math. 1186.
Arnold, L., Kliemann, W., and Oeljeklaus, E. (1986). Lyapunov exponents of linear stochastic systems. In Arnold, L., and Wihstutz, V. (eds.), Lyapunov exponents.Lect. Notes Math. 1186, 85–128.
Bougerol, P., and Lacroix, J. (1985).Products of Random Matrices with Applications to Schrödinger operators, Birkhäuser, Boston.
Boxler, P. (1988). Stochastische Zentrumsmannigfaltigkeiten. PhD thesis, Bremen University, Bremen.
Crauel, H. (1986). Lyapunov exponents and invariant measures of stochastic systems on manifolds. In Arnold, L., and Wihstutz, V. (eds.), Lyapunov exponents.Lect. Notes Math. 1186.
Crauel, H. (1988). Random dynamical systems: positivity of Lyapunov exponents, and Markov systems. PhD thesis, Bremen University, Bremen, 1987. Report 175, Institute for Dynamical Systems, Bremen University, Bremen.
Deimling, K. (1985).Nonlinear Functional Analysis, Springer, Berlin.
Guivarc'h, Y., and Raugi, A. (1985). Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence.Z. Wahrscheinlichkeitstheor. Verw. Geb. 69, 187–242.
Johnson, R., and Moser, J. (1982). The rotation number for almost periodic potentials.Commun. Math. Phys. 84, 403–438.
Oseledec, V. I. (1968). A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems.Trans. Moscow Math. Soc. 19, 197–231.
Ruelle, D. (1979). Ergodic theory of differentiable dynamical systems.Publ. Math. IHES 50, 27–58.
San Martin, L. (1988a). Rotation numbers of differential equations: a framework in the linear case. Relatório Técnico 05/88, Instituto de Matemática, UNICAMP, Campinas, SP, Brazil.
San Martin, L. (1988b). Rotation numbers in higher dimensions. Report 199, Institute for Dynamical Systems, Bremen University, Bremen.
Sell, G. (1978). The structure of a flow in the vicinity of an almost periodic motion.J. Differential Equations 27, 359–393.
Virtser, A. D. (1984). On the simplicity of the spectrum of the Lyapunov characteristic indices of a product of random matrices.Theory Probab. Appl. 28, 122–135.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Arnold, L., Martin, L.S. A multiplicative ergodic theorem for rotation numbers. J Dyn Diff Equat 1, 95–119 (1989). https://doi.org/10.1007/BF01048792
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01048792