Morse Decompositions on Flag Bundles and Uniform Bounds of Cocycles

Consider a principal bundle Q→ X Q → X with structural group G , which we assume to be a semi-simple or reductive Lie group. A flow φ ₜ ϕ t of automorphisms of Q induces a flow on the associated bundle E=Q× GF E = Q × G F , where F F is the maximal flag bundle of G . If we consider an ergodic probability measure on the compact base space X , this flow induces two decompositions of the flag bundles associated to Q : a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The polar decomposition of G=K( cl A⁺) K G = K cl A + K yields a map a⁺(n,x)∈ clA⁺ a + ( n , x ) ∈ cl A + over the flow induced on X . In this paper we prove a condition, called uniformity condition , which relates the Morse flag type Θ Mo Θ Mo to uniform lower bounds of the cocycle aα⁺(n,x) a α + ( n , x ) , x∈ X x ∈ X .