22 research outputs found
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 is the
global attractor associated with a dissipative partial differential equation in
a real Hilbert space and the set has finite Assouad dimension ,
then there is an ordinary differential equation in , with , 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 , where is a homeomorphism
from into
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
Continuous selections of multivalued mappings
This survey covers in our opinion the most important results in the theory of
continuous selections of multivalued mappings (approximately) from 2002 through
2012. It extends and continues our previous such survey which appeared in
Recent Progress in General Topology, II, which was published in 2002. In
comparison, our present survey considers more restricted and specific areas of
mathematics. Note that we do not consider the theory of selectors (i.e.
continuous choices of elements from subsets of topological spaces) since this
topics is covered by another survey in this volume
Unstable attractors in manifolds
Assume that K is a compact attractor with basin of attraction A(K) for some continuous flow phi 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 Poincare 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