Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)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.33412471273Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESP [06/60031-3, 07/06896-5]CNPq [305513/2003-6]