Let X be a symmetric space of noncompact type and Γa lattice in the isometry group of X. We study the distribution of orbits of Γacting on the symmetric space X and its geometric boundary X(\infty). More precisely, for any y in X and b in X(\infty), we investigate the distribution of the set {(yγ,bγ^{-1}):γ\inΓ} in X\times X(\infty). It is proved, in particular, that the orbits of Γin the Furstenberg boundary are equidistributed, and that the orbits of Γin X are equidistributed in ``sectors'' defined with respect to a Cartan decomposition. We also discuss an application to the Patterson-Sullivan theory. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces.
We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a-priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply …
Furstenberg has associated to every topological group $G$ a universal boundary $\partial(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\partial(G,H)$. In the case of discrete groups, an equivalent notion was introduced by Bearden--Kalantar as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $Δ(G,H)$, namely the s…
Let $G$ be a real linear semisimple algebraic group without compact factors and $Γ$ a Zariski dense subgroup of $G$. In this paper, we use a probabilistic counting in order to study the asymptotic properties of $Γ$ acting on the Furstenberg boundary of $G$. First, we show that the $K$ components of the elements of $Γ$ in the KAK decomposition of $G$ become asymptotically independent. This result is an analog of a result of Gorodnik-Oh in the context of the Archimedean counting. Then, we give a new proof of a result of Guivarc'h concerning the positivity of the Hausdorff dimension of the unique…
Let $μ$ be a measure on $SL_{2}(\mathbb{R})$ generating a non-compact and totally irreducible subgroup, let $χ>0$ denote its Lyapunov exponent, and let $ν$ be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if $μ$ is supported on finitely many matrices with algebraic entries, then \[ \dimν=\min\{1,\frac{h_{\textrm{RW}}(μ)}{2χ}\} \] where $h_{\textrm{RW}}(μ)$ is the random walk entropy of $μ$, and $\dim$ denotes pointwise dimension. In particular, for every $δ>0$, there is a neighborhood $U$ of the identity in $SL_{2}(\mathbb{R})$ such that i…
Let $Γ$ be a countable discrete group, $H$ a lcsc totally disconnected group and $ρ: Γ\rightarrow H$ a homomorphism with dense image. We develop a general and explicit technique which provides, for every compact open subgroup $L < H$ and bi-$L$-invariant probability measure $θ$ on $H$, a Furstenberg discretization $τ$ of $θ$ such that the Poisson boundary of $(H,θ)$ is a $τ$-boundary. Among other things, this technique allows us to construct examples of finitely supported random walks on certain lamplighter groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are prime, but no…
Equationally compact subgroups of countable groups were introduced by Banaschewski. For all known cases the orbit closure of such a subgroup is a countable subset in the space of subgroups and has finite Cantor-Bendixson rank. We show that there exists a finitely generated group $Γ$ such that for any countable ordinal $α$ we have an equationally compact subgroup $H\subset Γ$ for which the Cantor-Bendixson rank of the orbit closure of $H$ equals to $α+2$. Then we give an explicite construction of continuum many equationally compact subgroups of $Γ$ such that the associated ergodic Bernoulli shi…
We show recurrent phenomena for orbits of groups of local complex analytic diffeomorphisms that have a certain subgroup or image by a morphism of groups that is non-virtually solvable. In particular we prove that a non-virtually solvable subgroup of local biholomorphisms has always recurrent orbits, i.e. there exists an orbit contained in its set of limit points.
Somewhere dense orbit of abelian subgroup of diffeomorphisms maps acting on C^n
本文给出了C ^n中任意一个自同态李群G的交换子群G在C ^n中有某处稠密轨道G（x），x的一个刻划：G（x）在C ^n中某处稠密当且仅当存在f_{1}，.，f_{2n +1 in exp ^{-1}（G）使得f_{2n+1} in vect（f_{1}，...，f_{2n}）和Z. f_{1}（x）+.+ z. f_{2n+1}（x）是C ^n的稠密子群，其中向量（f_{1}，.，f_{2n}）是由f_{1}生成的R上的向量空间，.，f_{2n}。
We estimate the frequency of polynomial iterations which falls in a given multiplicative subgroup of a finite field of $p$ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $N$ elements in an orbit. We derive these from more general results about sequences of compositions or a fixed set of polynomials.
We discuss the local dynamics of a subgroup of Diff(C^2,0) possessing locally discrete orbits as well as the structure of the recurrent set for more general groups. It is proved, in particular, that a subgroup of Diff(C^2,0) possessing locally discrete orbits must be virtually solvable. These results are of considerable interest in problems concerning integrable systems.
We prove that a discrete subgroup generated by two lattices in opposite minimal horospherical subgroups of $SL(3,\mathbb{C})$ is arithmetic and thus by a Borel and Harish-Chandra also a lattice. We follow the method and ideas used by Oh in \cite{Ref13}. There, Oh proves the same result for lattices in minimal horospherical subgroups of $SL(n,\mathbb{R})$. The method consists of studying the orbits of the generating lattices in the horospherical subgroups under the conjugation action of the commutator of the corresponding Levi subgroup in order to obtain a rational structure for $SL(3,\mathbb{C…
We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues E(X,T) of the minimal Cantor system (X,T) is a subgroup of the intersection I(X,T) of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated to (X,T) is trivial, the quotient group I(X,T)/E(X,T) is torsion free. We give examples with non trivial infinitesimal subgroups where this proper…
Let $G$ be a semiabelian variety defined over an algebraically closed field $K$, endowed with a rational self-map $Φ$. Let $α\in G(K)$ and let $Γ\subseteq G(K)$ be a finitely generated subgroup. We show that the set $\{n\in\mathbb{N}\colon Φ^n(α)\in Γ\}$ is a union of finitely many arithmetic progressions along with a set of Banach density equal to $0$. In addition, assuming $Φ$ is regular, we prove that the set $S$ must be finite.
We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.
