Hypergroups on two tori

We classify all hypergroup structures on two disjoint tori and on two disjoint real lines. We show that both spaces admit many hypergroup structures, and that all examples can be constructed from group cases by substitution of open subhypergroups together with certain deformations of the group convolutions.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

Tight contact structures on solid tori

In this paper we study properties of tight contact structures on solid tori. In particular we discuss ways of distinguishing two solid tori with tight contact structures. We also give examples of unusual tight contact structures on solid tori.

We prove the existence of a -valued and a -valued invariant of a closed solid torus. We call them the self-linking number and the rotation number respectively. We then extend these definitions to the case of an open solid torus. We show that these invariants exhibit certain monotonicity properties with respect to inclusion. We also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic.

At the same time we discuss various ways of constructing a tight contact structure on a solid torus. We then produce examples of solid tori with tight contact structures and calculate self-linking and rotation numbers for these tori. These examples show that the invariants we defined do not give a complete classification of tight contact structure on open solid tori.

At the end, we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite-sheeted cover of that solid torus is no longer tight. This answers negatively a question asked by Eliashberg in 1990. We also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure, another example of unexpected behavior.

     

Centralizers of expanding maps on tori

We prove here that the elements of an open and dense subset of expanding maps on tori have trivial centralizers; i.e., the maps commute only with their own powers.The author was partially supported by CNPq, Brazil.     

Holomorphic Cartan geometries on complex tori

In [6], it was asked whether all flat holomorphic Cartan geometries $(G,H)$ on a complex torus are translation invariant. We answer this affirmatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type $(G,H)$, with G a complex affine Lie group, on any complex torus is translation invariant.     

Constructing center-stable tori

We show that certain derived-from-Anosov diffeomorphisms on the 2-torus may be realized as the dynamics on a center-stable or center-unstable torus of a 3-dimensional strongly partially hyperbolic system. We also construct examples of center-stable and center-unstable tori in higher dimensions.     

Wigner measures supported on weak KAM tori

In the setting of the Weyl quantization on the flat torus \(\mathbb{T}^n \) , we exhibit a class of wave functions with uniquely associated Wigner probability measure, invariant under the Hamiltonian dynamics and with support contained in weak KAM tori in phase space. These sets are the graphs of Lipschitz-continuous weak KAM solutions of negative type of the stationary Hamilton-Jacobi equation. Such Wigner measures are, in fact, given by the Legendre transform of Mather’s minimal probability measures.     

Circulant graphs and tessellations on flat tori

Circulant graphs are characterized here as quotient lattices, which are realized as vertices connected by a knot on a k-dimensional flat torus tessellated by hypercubes or hyperparallelotopes. Via this approach we present geometric interpretations for a bound on the diameter of a circulant graph, derive new bounds for the genus of a class of circulant graphs and establish connections with spherical codes and perfect codes in Lee spaces.     

Normalizers of maximal tori

Normalizers and p-normalizers of maximal tori in p-compact groups can be characterized by the Euler characteristic of the associated homogeneous spaces. Applied to centralizers of elementary abelian p-groups these criteria show that the normalizer of a maximal torus of the centralizer is given by the centralizer of a preferred homomorphism to the normalizer of the maximal torus; i.e. that “normalizer” commutes with “centralizer”. Received April 1, 1995; in final form August 11, 1997     

Motion planning in tori

Let X be a subcomplex of the standard CW-decomposition of then-dimensional torus. We exhibit an explicit optimal motion planningalgorithm for X. This construction is used to calculate thetopological complexity of complements of general position arrangementsand Eilenberg–Mac Lane spaces associated to right-angledArtin groups.     

Circulant graphs and tessellations on flat tori

Circulant graphs are characterized here as quotient lattices, which are realized as vertices connected by a knot on a k-dimensional flat torus tessellated by hypercubes or hyperparallelotopes. Via this approach we present geometric interpretations for a bound on the diameter of a circulant graph, derive new bounds for the genus of a class of circulant graphs and establish connections with spherical codes and perfect codes in Lee spaces.