EMBEDDING OF GLOBAL ATTRACTORS AND THEIR DYNAMICS

Suppose that A is the global attractor associated with a dissipative dynamical system on a Hilbert space H. If the set A - A has finite Assouad dimension d, then for any m > d there are linear homeomorphisms L : A -> Rm+1 such that LA is a cellular subset of Rm+1 and L-1 is log-Lipschitz (i.e. Lipschitz to within logarithmic corrections). We give a relatively simple proof that a compact subset X of R-k is the global attractor of some smooth ordinary differential equation on R-k if and only if it is cellular, and hence we obtain a dynamical system on R-k for which LA is the global attractor. However, LA consists entirely of stationary points. In order for the dynamics on LA to reproduce those on LA we need to make an additional assumption, namely that the dynamics restricted to A are generated by a log-Lipschitz continuous vector field (this appears overly restrictive when H is infinite-dimensional, but is clearly satisfied when the initial dynamical system is generated by a Lipschitz ordinary differential equation on R-N). Given this we can construct an ordinary differential equation in some R-k (where k is determined by d and alpha) that has unique solutions and reproduces the dynamics on A. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor chi arbitrarily close to LA.