### Some remarks on the random walk on finite groups

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We prove density modulo $1$ of the sets of the form $$\{{\mu}^{m}{\lambda}^{n}\xi +{r}_{m}:n,m\in \mathbb{N}\},$$ where $\lambda ,\mu \in \mathbb{R}$ is a pair of rationally independent algebraic integers of degree $d\ge 2,$ satisfying some additional assumptions, $\xi \ne 0,$ and ${r}_{m}$ is any sequence of real numbers.

We obtain upper and lower estimates for the Green function for a second order noncoercive differential operator on a homogeneous manifold of negative curvature.

In this survey article we start from the famous Furstenberg theorem on non-lacunary semigroups of integers, and next we present its generalizations and some related results.

Let $X$ be the quotient group of the $S$-adele ring of an algebraic number field by the discrete group of $S$-integers. Given a probability measure $\mu $ on ${X}^{d}$ and an endomorphism $T$ of ${X}^{d}$, we consider the relation between uniform distribution of the sequence ${T}^{n}\mathbf{x}$ for $\mu $-almost all $\mathbf{x}\in {X}^{d}$ and the behavior of $\mu $ relative to the translations by some rational subgroups of ${X}^{d}$. The main result of this note is an extension of the corresponding result for the $d$-dimensional torus ${\mathbb{T}}^{d}$ due to B. Host.

We obtain an estimate for the Poisson kernel for the class of second order left-invariant differential operators on higher rank NA groups.

We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group N and A = ℝ⁺. We prove that if F is harmonic and satisfies some growth condition then F has an asymptotic expansion as a → 0 with coefficients from 𝓓'(N). Then we single out a set of at most two of these coefficients which determine F. Then using asymptotic expansions we are able to prove some theorems...

Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to ${\mathbb{R}}^{k}$, k>1. We consider a class of second order left-invariant differential operators on S of the form ${\mathcal{L}}_{\alpha}={L}^{a}+{\Delta}_{\alpha}$, where $\alpha \in {\mathbb{R}}^{k}$, and for each $a\in {\mathbb{R}}^{k},{L}^{a}$ is left-invariant second order differential operator on N and ${\Delta}_{\alpha}=\Delta -\u27e8\alpha ,\nabla \u27e9$, where Δ is the usual Laplacian on ${\mathbb{R}}^{k}$. Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively) we obtain an...

Let Γ be a subsemigroup of G = GL(d,ℝ), d > 1. We assume that the action of Γ on ${\mathbb{R}}^{d}$ is strongly irreducible and that Γ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of Γ on ${\mathbb{R}}^{d}$ at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space ${\mathbb{P}}^{d-1}$. In the case where Γ is a subsemigroup of GL(d,ℝ) ∩ M(d,ℤ) and Γ has the above properties, we deduce that the Γ-orbits...

In this paper we treat noncoercive operators on simply connected manifolds of negative curvature.

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