The fundamental inequality of Guivarc'h relates the entropy and the drift of random walks on
groups. It is strict if and only if the random walk does not behave like the uniform measure on …

We give a general version of the Birkhoff ergodic theorem for functions taking values in non-
positively curved spaces. In this setting, the notion of a Birkhoff sum is replaced by that of a …

A general fixed point theorem for isometries in terms of metric functionals is proved under
the assumption of the existence of a conical bicombing. It is new for isometries of convex …

In this paper we study asymptotic properties of symmetric and nondegenerate random walks
on transient hyperbolic groups. We prove a central limit theorem and a law of iterated …

We consider dynamical and geometrical aspects of deep learning. For many standard
choices of layer maps we display semi-invariant metrics which quantify differences between …

In 1921, George Pólya published a short article [110] posing the following problem. Imagine
a traveller on an infinite regular grid of roads—an infinite Manhattan, without Broadway …

This paper discusses a general method for spectral type theorems using metric spaces
instead of vector spaces. Advantages of this approach are that it applies to genuinely non …

The notion of metric compactification was introduced by Gromov and later rediscovered by
Rieffel. It has been mainly studied on proper geodesic metric spaces. We present here a …

Thurston obtained a classification of individual surface homeomorphisms via the dynamics
of the corresponding mapping class elements on Teichmüller space. In this paper, we …

Consider a closed surface $ M $ with negative Euler characteristic, and an admissible
probability measure on the fundamental group of $ M $ with finite first moment …