Abstract. Assume that K is a compact attractor with basin of attraction A(K) for some continuous flow ϕ in a space M . Stable attractors are very well known, but otherwise (without the stability assumption) the situation can be extremely wild. In this paper we consider the class of attractors with no external explosions, where a mild form of instability is allowed.After obtaining a simple description of the trajectories in A(K) − K we study how K sits in A(K) by performing an analysis of the Poincaré polynomial of the pair (A(K), K). In case M is a surface we obtain a nice geometric characterization of attractors with no external explosions, as well as a converse to the well known fact that the inclusion of a stable attractor in its basin of attraction is a shape equivalence. Finally, we explore the strong relations which exist between the shape (in the sense of Borsuk) of K and the shape (in the intuitive sense) of the whole phase space M , much in the spirit of the Morse-Conley theory.

As a 1st step to understand how complicated attractors for dynamical systems can be, one may consider the following realizability problem: given a continuum $K \subseteq \mathbb{R}^3$, decide when $K$ can be realized as an attractor for a homeomorphism of $\mathbb{R}^3$. In this paper we introduce toroidal sets as those continua $K \subseteq \mathbb{R}^3$ that have a neighbourhood basis comprised of solid tori and, generalizing the classical notion of genus of a knot, give a natural definition of the genus of toroidal sets and study some of its properties. Using these tools we exhibit knots and solenoids for which the answer to the realizability problem stated above is negative.

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Abstract. 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 → R m+1 such that LA is a cellular subset of R m+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 α) 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 X arbitrarily close to LA.