We establish a maximin characterisation of the linear escape rate of the orbits of a non-expansive mapping on a complete (hemi-)metric space, under a mild form of Busemann's non-positive curvature condition (we require a distinguished family of geodesics with a common origin to satisfy a convexity inequality). This characterisation, which involves horofunctions, generalises the Collatz–Wielandt characterisation of the spectral radius of a non-negative matrix. It yields as corollaries a theorem of Kohlberg and Neyman (1981), concerning non-expansive maps in Banach spaces, a variant of a Denjoy–Wolff type theorem of Karlsson (2001), together with a refinement of a theorem of Gunawardena and Walsh (2003), concerning order-preserving positively homogeneous self-maps of symmetric cones. An application to zero-sum stochastic games is also given.

We study different aspects of the connections between local theory of Banach spaces and the problem of the extension of bilinear forms from subspaces of Banach spaces. Among other results, we prove that if X is not a Hilbert space then one may find a subspace of X for which there is no Aron–Berner extension. We also obtain that the extension of bilinear forms from all the subspaces of a given X forces such X to contain no uniform copies of ℓpn for p ∈ [1, 2). In particular, X must have type 2 − ϵ for every ϵ > 0. Also, we show that the bilinear version of the Lindenstrauss–Pełczyński and Johnson–Zippin theorems fail. We will then consider the notion of locally α-complemented subspace for a reasonable tensor norm α, and study the connections between α-local complementation and the extendability of α*-integral operators.

In this paper we analyse the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete noncompact case. In that case, our approach is based on the use of a generalized version of the Omori–Yau maximum principle for trace type differential operators, recently given by the authors in [3].

We construct a non-doubling measure on the real line, all tangent measures of which are equivalent to Lebesgue measure.

Unfortunately, there are two inaccuracies in the argument of [CLS]. First, the statements of Lemmas 3, 4, 6, and 7 of [CLS] hold only under the additional condition gcd(m, ME) = 1 for some integer ME ≥ 1 depending only on E. Second, the divisibility condition (3·6) in [CLS] implies that tb(ℓ) | nE(p)−1 (rather than tb(ℓ) | nE(p), as it was erroneously claimed on p. 519 in [CLS]). In particular, instead of the divisibility ℓtb(ℓ) | nE(p) (see the last displayed formula on p. 519 in [CLS]), we conclude that for every prime ℓ | L there is an integer aℓ such that (0.1) However, the final result is correct and can easily be recovered. To do so, we remark that under the condition gcd(m,ME) =1, we have full analogues of Lemmas 6, 7, 9, and 10 of [CLS] for the function Π(x;m,a) defined as the number of primes p ≤ x with nE(p) ≡ a (mod m) (rather than just for Π(x;m) = Π(x;m,0) as in [CLS]). Define ρ*(n) as the largest square-free divisor of n which is relatively prime to ME. We then derive from (0.1) above that Therefore (0.2) Since we see that (0.2) above implies the bound (3·7) from [CLS], and the result now follows without any further changes.

We study some properties of Lipschitz exact Lagrangian manifolds isotopic to the zero section. We prove that if such a manifold is invariant under an optical Hamiltonian, then it must be a Lipschitz graph. This extends a recent result of Marie–Claude Arnaud. We also obtain a new geometric description of the Mañé–Mather invariant set.

Improving on estimates of Erdős, Halász and Ruzsa, we provide new upper and lower bounds for the concentration function of the limit law of certain additive arithmetic functions under hypotheses involving only their average behaviour on the primes. In particular we partially confirm a conjecture of Erdős and Kátai. The upper bound is derived via a reappraisal of the method of Diamond and Rhoads, resting upon the theory of functions with bounded mean oscillation.

Suite à une remarque de Dimitris Koukoulopoulos, que nous prenons plaisir à remercier ici, nous modifions légèrement l'énoncé et la démonstration du Corollaire 1.4, de façon à justifier la remarque (ii).