On the Continuity of Global Attractors

Let Λ be a complete metric space, and let {Sλ(·) : λ ∈ Λ} be a parametrised family of semigroups with global attractors Aλ. We assume that there exists a fixed bounded set D such that Aλ ⊂ D for every λ ∈ Λ. By viewing the attractors as the limit as t → ∞ of the sets Sλ(t)D, we give simple proofs of the equivalence of ‘equi-attraction’ to continuity (when this convergence is uniform in λ) and show that the attractors Aλ are continuous in λ at a residual set of parameters in the sense of Baire Category (when the convergence is only pointwise). 1. Global attractors The global attractor of a dynamical system is the unique compact invariant set that attracts the trajectories starting in any bounded set at a uniform rate. Introduced by Billotti & LaSalle [3], they have been the subject of much research since the mid-1980s, and form the central topic of a number of monographs, including Babin & Vishik [1], Hale [9], Ladyzhenskaya [13], Robinson [16], and Temam [18]. The standard theory incorporates existence results [3], upper semicontinuity [10], and bounds on the attractor dimension [7]. Global attractors exist for many infinitedimensional models [18], with familiar low-dimensional ODE models such as the Lorenz equations providing a testing ground for the general theory [8]. While upper semicontinuity with respect to perturbations is easy to prove, lower semicontinuity (and hence full continuity) is more delicate, requiring structural assumptions on the attractor or the assumption of a uniform attraction rate. However, Babin & Pilyugin [2] proved that the global attractor of a parametrised set of semigroups is continuous at a residual set of parameters, by taking advantage of the known upper semicontinuity and then using the fact that upper semicontinuous functions are continuous on a residual set. Here we reprove results on equi-attraction and residual continuity in a more direct way, which also serves to demonstrate more clearly why these results are true. Given equi-attraction the attractor is the uniform limit of a sequence of continuous functions, and hence continuous (the converse requires a generalised version of Dini’s Theorem); more generally, it is the pointwise limit of a sequence of continuous functions, i.e. a ‘Baire one’ function, and therefore the set of continuity points forms a residual set. 2000 Mathematics Subject Classification. Primary 35B41.