Embedding of global attractors and their dynamics

Abstract

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