Related papers: Several remarks on Pascal automorphism and infinit…

In this paper, we study random walks on groups that contain superlinear divergent geodesics, in the line of thoughts of Goldsborough-Sisto. The existence of a superlinear divergent geodesic is a quasi-isometry invariant which allows us to…

Let X be a finite Abelian group, xi_i, i=1,2,...,n,n>1, be independent random variables with values in X and distributions mu_i. Let alpha_{ij},i,j=1,2,...,n, be automorphisms of X. We prove that the independence of n linear forms…

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by…

We introduce an adic (Bratteli-Vershik) dynamical system based on a diagram whose path counts from the root are the Delannoy numbers. We identify the ergodic invariant measures, prove total ergodicity for each of them, and initiate the…

We consider the activated random walk particle system, a model of self-organized criticality, on $\mathbb{Z}$ with i.i.d.-Bernoulli initial configuration. We show that at subcritical density, the system's odometer function, which counts the…

We show that the ergodicity of an aperiodic automorphism of a Lebesgue space is equivalent to the continuity of a certain map on a metric Boolean algebra. A related characterization is also presented for periodic and totally ergodic…

This is an expository article on properties of actions on Lie groups by subgroups of their automorphism groups. After recalling various results on the structure of the automorphism groups, we discuss actions with dense orbits, invariant and…

In this paper, we discuss a group-theoretical generalization of the well-known Gauss formula involving the functionthat counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.

This paper defines, on the Galilean space-time, the group of asymptotically Euclidean transformations (AET), which are equivalent to Euclidean transformations at space-time infinity, and proposes a formulation of nonrelativistic quantum…

We consider isometric actions of lattices in semisimple algebraic groups on (possibly non-compact) homogeneous spaces with (possibly infinite) invariant Radon measure. We assume that the action has a dense orbit, and demonstrate two novel…

We develop an explicit geometric construction of automorphisms of finite fields arising from isogeny cycles. Let $k$ be a finite field, $E/k$ an elliptic curve, and $\ell$ an integer coprime to $\mathrm{char}(k)$. Let $\mathfrak{h}$ be an…

We realize the infinitesimal Abel-Jacobi map as a morphism of formal deformation theories, realized as a morphism in the homotopy category of differential graded Lie algebras. The whole construction is carried out in a general setting, of…

The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular…

We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary…

We give examples of rank-one transformations that are (weak) doubly ergodic and rigid (so all their cartesian products are conservative), but with non-ergodic $2$-fold cartesian product. We give conditions for rank-one infinite…

Previous work introduced two measure-conjugacy invariants: the $f$-invariant (for actions of free groups) and $\Sigma$-entropy (for actions of sofic groups). The purpose of this paper is to show that the $f$-invariant is a special case of…

Ergodic theory includes several notions of entropy for probability-preserving actions of countable groups. These include Kolmogorov--Sinai entropy based on F\o lner sequences for amenable groups, entropy defined using a random ordering of…

This survey is focused on the results related to topologies on the groups of transformations in ergodic theory, Borel, and Cantor dynamics. Various topological properties (density, connectedness, genericity) of these groups and their…

The theory of voltage graphs has become a standard tool in the study graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting…

In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by…