W. P. Thurston introduced closed σd-invariant laminations (where σd = z : S → S, d ≥ 2) as a tool in complex dynamics. He defined wandering triangles as triples T ⊂ S such that σ d (T ) consists of three distinct points for all n ≥ 0 and the convex hulls of all the sets σ d (T ) in the plane are pairwise disjoint, and proved that σ2 admits no wandering triangles. We show that for every d ≥ 3 th...

For a manifold $M$ with an integral closed 3-form $\omega$, we construct $PU(H)$-bundle and Lie groupoid over its total space, together curving in the sense of gerbes. If form is non-degenerate, furthermore give natural 2-algebra quasi-isomorphism from observables $(M,\omega)$ to weak symmetries above geometric structure, generalising prequantisation map Kostant Souriau.

Consider a form g(x1, . . . , xs) of degree d, having coefficients in the completion Fq((1/t)) of the field of fractions Fq(t) associated to the finite field Fq. We establish that whenever s > d, then the form g takes arbitrarily small values for non-zero arguments x ∈ Fq[t ]. We provide related results for problems involving distribution modulo Fq[t ], and analogous conclusions for quasi-algeb...