Simple polytopes without small separators,II: Thurston’s bound

We construct a diffeomorphism of T³ admitting any finite or countable number of physical measures with intermingled basins. The examples are partially hyperbolic with splitting TT³ = E ^cs ⊕ E ^u and can be made volume hyperbolic and topologically mixing.     

Examples and Counterexamples for the Perles Conjecture

   Abstract. The combinatorial structure of a d -dimensional simple convex polytope—as given, for example, by the set of the (d-1) -regular subgraphs of the facets—can be reconstructed from its abstract graph. However, no polynomial/ efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction exists. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by Micha Perles: ``The facet subgraphs of a simple d-polytope are exactly all the (d-1) -regular, connected, induced, non-separating subgraphs .' We present non-trivial classes of examples for the validity of the Perles conjecture: in particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes. On the other hand, we observe that for any 4-dimensional counterexample, the boundary of the (simplicial) dual polytope

A tight colored Tverberg theorem for maps to manifolds

We prove that any continuous map of an N-dimensional simplex ΔN with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of ΔN to the same point in M: For this we have to assume that N?(r−1)(d+1), no r vertices of ΔN get the same color, and our proof needs that r is a prime. A face of ΔN is a rainbow face if all vertices have different colors.This result is an extension of our recent “new colored Tverberg theorem”, the special case of M=Rd. It is also a generalization of Volovikov?s 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikov?s proof, as well as ours, works when r is a prime power.     

Projections of polytopes and the generalized baues conjecture

Associated with every projection π:P→π(P) of a polytopeP is a partially ordered set of all “locally coherent strings”: the families of proper faces ofP that project to valid subdivisions of π(P), partially ordered by the natural inclusion relation. The “Generalized Baues Conjecture” posed by Billeraet al. [4] asked whether this partially ordered set always has the homotopy type of a sphere of dimension dim(P—dim(π(P))?1. We show that this is true in the cases when dim(π(P))=1 (see[4]) and when dim(P)—dim(π(P))≤2, but fails in general. For an explicit counterexample we produce a nondegenerate projection of a five-dimensional, simplicial, 2-neighborly polytopeP with 10 vertices and 42 facets to a hexagon π(P)??². The construction of the counterexample is motivated by a geometric analysis of the relation between the fibers in an arbitrary projection of polytopes.     

Film transitions of receding contact lines

When a solid plate is withdrawn from a liquid bath, a receding contact line is formed where solid, liquid, and gas meet. Above a critical speed U_cr, a stationary contact line can no longer exist and the solid will eventually be covered completely by a liquid film. Here we show that the bifurcation diagram of this coating transition changes qualitatively, from discontinuous to continuous, when decreasing the inclination angle θ_p of the plate. We show that this effect is governed by the presence of capillary waves, illustrating that the large scale flow strongly effects the maximum speed of dewetting.     

IR and SFM study of PTCDA thin films on different substrates

FT-IR spectroscopy and SFM were used to investigate the growth of thin films of the organic semiconductor 3,4,9,10-perylenetetracarboxylicdianhydride (PTCDA) deposited by vacuum sublimation onto various substrates, i.e. Ag(111) layers on mica, KBr(100), mica, oxidized Si, and TiO₂ nanoparticles on Si. Layer thicknesses of PTCDA varied from 10 to 1500 nm.The anhydride vibrations of PTCDA differ for the used substrates, which can be connected to the orientation of the molecules relative to the substrate surface and the film morphology as detected in the SFM pictures.