On the set of wild points of attracting surfaces in R3

Suppose that a closed surface S⊆R3is an attractor, notne-cessarily global, for a discrete dynamical system. Assuming that its set of wild points Wis totally disconnected, we prove that (up to an ambient homeomorphism) it has to be con-tained in a straight line. As a corollary we show that there exist uncountably many different 2-spheres in R3 none of which can be realized as an attractor for a homeomorphism. Our techniques hinge on a quantity r(K)that can be de-fined for any compact set K⊆R3and is related to “how wildly” it sits in R3. We establish the topological results that (i)r(W) ≤r(S)and (ii) any totally disconnected set having a finite rmust be contained in a straight line (up to an ambient homeomorphism). The main result follows from these and the fact that attractors have a finite r.The author is supported by the Spanish Ministerio de Economía y Competitividad (grant MTM 2015-63612-P)

Embedding of global attractors and their dynamics

Using shape theory and the concept of cellularity, we show that if $A$ is the global attractor associated with a dissipative partial differential equation in a real Hilbert space $H$ and the set $A-A$ has finite Assouad dimension $d$, then there is an ordinary differential equation in ${\mathbb R}^{m+1}$, with $m >d$, 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 $LA$, where $L$ is a homeomorphism from $A$ into ${\mathbb R}^{m+1}$

Universal bounds on the entropy of toroidal attractors

A toroidal set is a compactum $K \subseteq \mathbb{R}^3$ which has a neighbourhood basis of solid tori. We study the topological entropy of toroidal attractors $K$, bounding it from below in terms of purely topological properties of $K$. In particular, we show that for a toroidal set $K$, either any smooth attracting dynamics on $K$ has an entropy at least $\log 2$, or (up to continuation) $K$ admits smooth attracting dynamics which are stationary (hence with a zero entropy)

On the components of the unstable set of isolated invariant sets

The aim of this note is to shed some light on the topological structure of the unstable set of an isolated invariant set K. We give a bound on the number of essential quasicomponents of the unstable set of K in terms of the homological Conley index of K. The proof relies on an explicit pairing between Čech homology classes and Alexander–Spanier cohomology classes that takes the form of an integral.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEMinisterio de Ciencia, Innovación y Universidadespu