MULTIPLICATIVE ERGODIC THEOREM ON FLAG BUNDLES OF SEMI-SIMPLE LIE GROUPS

Author(s)

Alves, Luciana A. ^[1] ; San Martin, Luiz A. B. ^[2]

Affiliation

^[1] Univ Fed Uberlandia, Fac Matemat, BR-38408100 Uberlandia, MG - Brazil

^[2] Imecc Unicamp, Dept Matemat, Campinas, SP - Brazil

• Document type: Journal article
• Source: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS; v. 33, n. 4, p. 1247-1273, APR 2013.
• Web of Science Citations: 2

Abstract

Let Q -> X be a principal bundle having as structural group G a reductive Lie group in the Harish-Chandra class that includes the case when G is semi-simple with finite center. A semiflow phi(k) of endomorphisms of Q induces a semiflow psi(k) on the associated bundle E = Q x(G) F, where F is the maximal flag bundle of G. The A-component of the Iwasawa decomposition G = KAN yields an additive vector valued cocycle a (k, xi), xi is an element of E, over psi(k) with values in the Lie algebra a of A. We prove the Multiplicative Ergodic Theorem of Oseledets for this cocycle: If nu is a probability measure invariant by the semiflow on X then the a-Lyapunov exponent lambda (xi) = lim 1/ka (k, xi) exists for every xi on the fibers above a set of full nu-measure. The level sets of lambda (.) on the fibers are described in algebraic terms. When phi(k) is a flow the description of the level sets is sharpened. We relate the cocycle a (k, xi) with the Lyapunov exponents of a linear flow on a vector bundle and other growth rates. (AU)

• FAPESP's process: 07/06896-5 - Geometry of control, dynamical and stochastic systems
• Grantee: Luiz Antonio Barrera San Martin
• Support Opportunities: Research Projects - Thematic Grants