We consider the group of isotopy classes of automorphisms of the 3-sphere that preserve a spatial graph or a handlebody-knot embedded in it. We prove that the group is finitely presented for an arbitrary spatial graph or a reducible handlebody-knot of genus two. We also prove that the groups for “most” irreducible genus two handlebody-knots are finite.

The notion of an exponential contraction is only one among many possible rates of contraction of a nonautonomous system, while for an autonomous system all contractions are exponential. We consider the notion of an L1 contraction that includes exponential contractions as a very particular case and that is naturally adapted to the variation-of-parameters formula. Both for discrete and continuous time, we show that under very general assumptions the notion of an L1 contraction persists under sufficiently small linear and nonlinear perturbations, also maintaining the type of stability. As a natural development, we establish a version of the Grobman–Hartman theorem for nonlinear perturbations of an L1 contraction.

The well-known theorems of Khintchine and Jarník in metric Diophantine approximation provide a comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this paper we develop the theory for planar curves by completing the theory in the case of parabola. This represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.

We analyse the Gottlieb groups of function spaces. Our results lead to explicit decompositions of the Gottlieb groups of many function spaces map(X, Y)—including the (iterated) free loop space of Y—directly in terms of the Gottlieb groups of Y. More generally, we give explicit decompositions of the generalised Gottlieb groups of map(X, Y) directly in terms of generalised Gottlieb groups of Y. Particular cases of our results relate to the torus homotopy groups of Fox. We draw some consequences for the classification of T-spaces and G-spaces. For X, Y finite and Y simply connected, we give a formula for the ranks of the Gottlieb groups of map(X, Y) in terms of the Betti numbers of X and the ranks of the Gottlieb groups of Y. Under these hypotheses, the Gottlieb groups of map(X, Y) are finite groups in all but finitely many degrees.

We show that a torus knot which is not 2-bridge has a unique irreducible bridge splitting of positive genus.

We refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(Fn). This gives restrictions to actions of Aut(Fn) on CAT(0) spaces.

Let HX be the trigraded Hilbert function of a set X of reduced points in $\mathbb{P}$1 × $\mathbb{P}$1 × $\mathbb{P}$1. We show how to extract some geometric information about X from HX. This paper generalises a similar result of Giuffrida, Maggioni and Ragusa about sets of points in $\mathbb{P}$1 × $\mathbb{P}$1.

Consider a network linking the points of a rate-1 Poisson point process on the plane. Write Ψave(s) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most s times the Euclidean distance. We give upper and lower bounds on the function Ψave(s), and on the analogous “worst-case” function Ψworst(s) where the point configuration is arbitrary subject to average density one per unit area. Our bounds are numerically crude, but raise the question of whether there is an exponent α such that each function has Ψ(s) ≍ (s − 1)−α as s ↓ 1.

Using class field theory we prove an explicit result of André–Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C})$ are the singular moduli, while the special points of $\mathbb{G}_m(\mathbb{C})$ are defined to be the roots of unity.

Let K be a function field over an algebraically closed field k of characteristic 0, let ϕ ∈ K(z) be a rational function of degree at least equal to 2 for which there is no point at which ϕ is totally ramified and let α ∈ K. We show that for all but finitely many pairs (m, n) ∈ $\mathbb{Z}$⩾0 × $\mathbb{N}$ there exists a place $\mathfrak{p}$ of K such that the point α has preperiod m and minimum period n under the action of ϕ. This answers a conjecture made by Ingram–Silverman [13] and Faber–Granville [8]. We prove a similar result, under suitable modification, also when ϕ has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple (c0, . . ., cd−2) ∈ kd−1 and for almost all pairs (mi, ni) ∈ $\mathbb{Z}$⩾0 × $\mathbb{N}$ for i = 0, . . ., d − 2, there exists a polynomial f ∈ k[z] of degree d in normal form such that for each i = 0, . . ., d − 2, the point ci has preperiod mi and minimum period ni under the action of f.

In the article [1] we claimed a strict inequality n(n – 1) < 4g(S) for an abelian surface link S of rank n (Theorem 2.1).