Open AccessMultiplicative ergodic theorem on flag bundles of semi-simple Lie groups
Author(s) -
Luciana Aparecida Alves,
Luiz A. B. San Martín
Publication year - 2013
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2013.33.1247
Subject(s) - ergodic theory , multiplicative function , mathematics , lie group , center (category theory) , combinatorics , flag (linear algebra) , lie algebra , physics , pure mathematics , mathematical analysis , algebra over a field , crystallography , chemistry
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