We give an example of a sequence of positive harmonic functions on \mathbb{Z}^{d} , d\geq 2 , that converges pointwise to a non-harmonic function.
We provide a proof of the Baum-Connes conjecture (with no coefficients) for word hyperbolic groups, based on the Green metric, a metric adapted to random walks.
We provide a proof of the Baum–Connes conjecture (with no coefficients) for word hyperbolic groups, based on the Green metric, a metric adapted to random walks. Résumé Nous proposons une démonstration de la conjecture de Baum–Connes (sans coefficients) pour les groupes hyperboliques en utilisant la distance de Green, une distance adaptée à l’étude des marches aléatoires.
We study the dynamics of complex algebraic families of maps on \mathbb{P}^{N} , and the geometry of their preperiodic points. The goal of this article is to formulate a conjectural characterization of the subvarieties of S \times\mathbb{P}^{N} containing a Zariski-dense set of preperiodic points, where the parameter space S is a quasi-projective complex algebraic variety; the characterization is given in terms of the non-vanishing of a power of the invariant Green current associated to the family of maps. This conjectural characterization is inspired by and generalizes the relative Manin–Mumford conjecture for families of abelian varieties, recently proved by Gao and Habegger, and it includes as special cases the Manin–Mumford conjecture (theorem of Raynaud) and the dynamical Manin–Mumford conjecture (posed by Ghioca, Tucker, and Zhang). We provide examples where the equivalence is known to hold, and we show that many recent results can be viewed as special cases. Finally, we give the proof of one implication in the conjectural characterization.
For a finite metric graph X=(V,E,ℓ), where V is endowed with the shortest path metric, we consider the transportation cost problem associated with the distance d on V. Namely, for f a function with total sum 0 on V, write f=∑_a,b∈ VP(a,b)(δ_a-δ_b) where the transportation plan P satisfies P(a,b)≥ 0 for (a,b)∈ V× V. The cost of P is W(P):=∑_a,b∈ VP(a,b)d(a,b) and the transportation norm of f is f_TC=min_P W(P) where P runs over all transportation plans for f. In this semi-survey paper, we give short proofs for the following statements: 1)There always exists an optimal transportation plan supported in V_+× V_- where V_+={x∈ V: f(x)>0} and V_-={x∈ V: f(x)<0}. If X is a metric tree, we may moreover assume that this plan involves at most |Supp(f)|-1 transports. 2) There always exists an optimal transportation plan supported in the set of edges of X. 3) Better, there always exists an optimal transportation plan supported in some spanning tree of X. We use this to reprove known formulae for the transportation norm when X is either a tree or a cycle.
