Structural Stability And Topological Classification Of Continuous-Time Linear Hyperbolic Cocycles

. In this paper we study topological properties of continuoustime linear hyperbolic cocycles. Roughly speaking, two cocycles are called conjugate if there exists a random homeomorphism mapping their orbits into each other; a cocycle is called structurally stable if it is conjugate to every cocycle from a neighborhood of itself. We prove that any linear hyperbolic cocycle is structurally stable with respect to its Lyapunov norm for all sufficiently small values of the parameter a in the definition of the Lyapunov norm. Concerning the classification problem, in the deterministic case it is well known that two linear hyperbolic flows are topologically equivalent if they have stable subspaces of the same dimension, and hence there are d + 1 topological classes of d-dimensional linear hyperbolic flows (see Irwin [11], p. 86). In this paper we prove that two cocycles are conjugate if and only if they have stable subspaces of the same dimension and their restrictions to their stable subspaces..