Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes’s example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is generalized to compact, locally rigid, ultrametric spaces. The local isometry ty...

Free evolution for quantum particle in general ultrametric space is considered. We find that if mean zero wave packet is localized in some ball in the ultrametric space then its evolution remains localized in the same ball.

Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is generalized to compact, locally rigid, ultrametric spaces. The local isometry ty...

We discuss the interbasin kinetics approximation for random walk on a complex landscape. We show that for a generic landscape the corresponding model of interbasin kinetics is equivalent to an ultrametric diffusion, generated by an ultrametric pseudodifferential operator on the ultrametric space related to the tree of basins. The simplest example of ultrametric diffusion of this kind is describ...

— We study the geometry of the space of measures of a compact ultrametric space X , endowed with the L Wasserstein distance from optimal transportation. We show that the power p of this distance makes this Wasserstein space affinely isometric to a convex subset of l. As a consequence, it is connected by 1 p -Hölder arcs, but any α-Hölder arc with α > 1 p must be constant. This result is obtaine...

Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is generalized to compact, locally rigid, ultrametric spaces. The local isometry ty...

We investigate the continuous model theory of ultrametric spaces of diameter ≤ 1. There is no universal Polish ultrametric space of diameter 1; but there is a Polish ultrametric space, Umax, taking distances in Q∩[0, 1], which is universal for all such Polish ultrametric spaces. We show that in the continuous theory of Umax, nonforking is characterized by a stable independence relation, which i...

Ametric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces [1], we show that a countable ultrametric space embeds isometrically into an indivisible ultrametric metric space if and only if it does not contain a strictly increasing sequence of balls.

We introduce a continuous-time quantum walk on an ultrametric space corresponding to the set of p-adic integers and compute its time-averaged probability distribution. It is shown that localization occurs for any location of the ultrametric space for the walk. This result presents a striking contrast to the classical random walk case. Moreover we clarify a difference between the ultrametric spa...