SPECTRA OF DELAY EQUATIONS

We describe all possible forms of the Sacker–Sell spectrum for a nonautonomous linear delay equation. In particular, the spectrum may have infinitely many connected components either accumulating at minus infinity or at a finite value. We also show that to each connected component of the spectrum one can associate a subspace of initial conditions whose lower and upper Lyapunov exponents are contained in the connected component. These spaces generate the whole ambient space. Finally, we give a condition under which the Lyapunov exponents of a nonlinear perturbation of a linear delay equation belong to a connected component of the spectrum.