Invariant densities of delayed maps in the limit of large time delay

The marginal invariant density of chaotic attractors of scalar systems with time delayed feedback has an asymptotic form in the limit of large delay. It is well known that the dimension and the entropy of such attractors obey interesting scaling laws in this limit, but very little has been said about properties of the invariant density. We present general considerations, detailed analytical results in low order perturbation theory for a particular model, and numerics for understanding the asymptotic behavior of the projections of the invariant density. Our approach clarifies how the analytical properties of the model determine the behavior of the marginal invariant densities for large delay times.