Topological Classiication of Linear Hyperbolic Cocycles

In this paper linear hyperbolic cocycles are classiied by the relation of topological conjugacy. Roughly speaking, two linear cocycles are conjugate if there exists a homeomorphism which maps their tra-jectories into each other. The problem of classiication of discrete-time deterministic hyperbolic dynamical systems was investigated by Rob-bin (1972). He proved that there exist 4d classes of d-dimensional deterministic discrete hyperbolic dynamical systems. We obtain a criterion for topological conjugacy of two linear hyperbolic cocycles and show that the number of classes depends crucially on the ergodic properties of the metric dynamical system over which they are deened. Our result is a generalization of the deterministic theorem of Robbin.