  We develop the dichotomy spectrum for random dynamical system and demonstrate
its use in the characterization of pitchfork bifurcations for random dynamical
systems with additive noise.
  Crauel and Flandoli had shown earlier that adding noise to a system with a
deterministic pitchfork bifurcation yields a unique attracting random fixed
point with negative Lyapunov exponent throughout, thus "destroying" this
bifurcation. Indeed, we show that in this example the dynamics before and after
the underlying deterministic bifurcation point are topologically equivalent.
  However, in apparent paradox, we show that there is after all a qualitative
change in the random dynamics at the underlying deterministic bifurcation
point, characterized by the transition from a hyperbolic to a non-hyperbolic
dichotomy spectrum. This breakdown manifests itself also in the loss of uniform
attractivity, a loss of experimental observability of the Lyapunov exponent,
and a loss of equivalence under uniformly continuous topological conjugacies.
