arXiv subjects
Publications and source records attributed to Robbert Fokkink.
We study a one-parameter family of quadratic maps, which serves as a template for interior point methods. It is known that such methods can exhibit chaotic behavior, but this has been verified only for particular linear optimization problems. Our results indicate that this chaotic behavior is generic.
Blanc showed in his thesis that a compact minimal foliated space with a residual subset of 2-ended leaves can contain only 1 or 2 ended leaves. In this paper we give examples of compact minimal foliated spaces where a topologically generic leaf has 1 end, there is an uncountable set of leaves with 2 ends and a leaf with 2n ends, for a given n>1. The examples we present are weak solenoids, which allows us to represent the graph of the group action on the fibre as the inverse limit of finite coverings of a finite graph, which we call the Schreier continuum, which we use to obtain the result. While in certain cases the problem can be reduced to the study of a self-similar action of an automorphism group of a regular tree, our geometric technique is more general, as it applies to cases where the action is not self-similar.
Lind and Schmidt have shown that the homoclinic group of a cyclic $\Z^k$ algebraic dynamical system is isomorphic to the dual of the phase group. We show that this duality result is part of an exact sequence if $k=1$. The exact sequence is a well known algebraic object, which has been applied by Schmidt in his work on rigidity. We show that it can be derived from dynamical considerations only. The constructions naturally lead to an almost 1-1-coding of certain Pisot automorphisms by their associated $β$-shift, generalizing similar results for Pisot automorphisms of the torus.
Solenoids are inverse limit spaces over regular covering maps of closed manifolds. M.C. McCord has shown that solenoids are topologically homogeneous and that they are principal bundles with a profinite structure group. We show that if a solenoid is bihomogeneous, then its structure group contains an open abelian subgroup. This leads to new examples of homogeneous continua that are not bihomogeneous.