Let $G$ be a Lie group, $Γ\subset G$ a discrete subgroup, $X=G/Γ$, and $f$ an affine map from $X$ to itself. We give conditions on a submanifold $Z$ of $X$ guaranteeing that the set of points $x\in X$ with $f$-trajectories avoiding $Z$ is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on $X$. This has applications to constructing exceptional geodesics on locally symmetric spaces, and to non-density of the set of values of certain functions at integer points.
Let $λ$ denote the probability Lebesgue measure on ${\mathbb T}^2$. For any $C^2$-Anosov diffeomorphism of the $2$-torus preserving $λ$ with measure-theoretic entropy equal to topological entropy, we show that the set of points with nondense orbits is hyperplane absolute winning (HAW). This generalizes the result in~\cite[Theorem~1.4]{T4} for $C^2$-expanding maps of the circle.
We show that set of points nondense under the $\times n$-map on the circle and dense for the geodesic flow under the induced map on the circle corresponding to the expanding horospherical subgroup has full Haudorff dimension. We also show the analogous result for toral automorphisms on the $2$-torus and a diagonal flow. Our results can be interpreted in number-theoretic terms: the set of well approximable numbers that are nondense under the $\times n$-map has full Hausdorff dimension. Similarly, the set of well approximable $2$-vectors that are nondense under a hyperbolic toral automorphism ha…
We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and nondense forward orbits under the other is a dense, uncountable set. The pair of maps can be noncommuting. We also show the same for pairs of $C^2$-Anosov diffeomorphisms on the $2$-torus. (The pairs must satisfy slight constraints.) Our main tools are the Baire Category theorem and a geometric construction that allows us to give a geometric characterization of the fractal that is the set of points with forward orbits that miss a certain open set.
We show in prime dimension that for two non-commuting totally irreducible toral automorphisms the set of points that equidistribute under the first map but have non-dense orbit under the second has full Hausdorff dimension. In non-prime dimension the argument fails only if the automorphisms have strong algebraic relations.
We will prove an S-arithmetic version of a theorem of Dani-Margulis on the convergence of ergodic averages of a given bounded continuous function, when the initial point is outside certain compact subsets of the singular set associated to the unipotent flow.
In this paper we introduce the notion of orbit equivalence for semigroup actions and the concept of generalized linear control system on smooth manifold. The main goal is to prove that, under certain conditions, the semigroup system of a generalized linear control system on a smooth manifold is orbit equivalent to the semigroup system of a linear control system on a homogeneous space.
Let $\G$ be a semisimple algebraic group defined over a number field $K$, $\te$ a maximal $K$-split torus of $\G$, $\mathcal{S}$ a finite set of valuations of $K$ containing the archimedean ones, $\OO$ the ring of $\mathcal{S}$-integers of $K$ and $K_\mathcal{S}$ the direct product of the completions $K_v, v \in \mathcal{S}$. Denote $G = \G(K_\mathcal{S})$, $T = \te(K_\mathcal{S})$ and $Γ= \G(\OO)$. Let $Tπ(g)$ be a locally divergent orbit for the action of $T$ on $G/Γ$ by left translations. We prove: ($1$) if $\# S = 2$ then the closure $\overline{Tπ(g)}$ is a union of finitely many $T$-orbit…
Let $(X,\mathfrak{B},μ)$ be a Borel probability space. Let $T_n: X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $ν$ be a probability measure on $X$ such that $\frac{1}{N}\sum_{n=1}^N (T_n)_\ast ν\rightarrow μ$ in the weak-$\ast$ topology. Under general conditions, we show that for $ν$ almost every $x\in X$, the measures $\frac{1}{N}\sum_{n=1}^N δ_{T_n x}$ get equidistributed towards $μ$ if $N$ is restricted to a set of full upper density. We present applications of these results to translates of closed orbits of Lie groups on homogeneous spaces. As a corollary, we prov…
Let $\G$ be a semisimple algebraic group over a number field $K$, $\mathcal{S}$ a finite set of places of $K$, $K_\mathcal{S}$ the direct product of the completions $K_v, v \in \mathcal{S}$, and $\OO$ the ring of $\mathcal{S}$-integers of $K$. Let $G = \G(K_\mathcal{S})$, $Γ= \G(\OO)$ and $π:G \rightarrow G/Γ$ the quotient map. We describe the closures of the locally divergent orbits ${Tπ(g)}$ %in $G/Γ$ where $T$ is a maximal $K_\mathcal{S}$-split torus in $G$. If $\# S = 2$ then the closure $\overline{Tπ(g)}$ is a finite union of $T$-orbits stratified in terms of parabolic subgroups of $\G \t…
Let $X = G/Γ$, where $G$ is a Lie group and $Γ$ is a lattice in $G$, and let $U$ be a subset of $X$ whose complement is compact. We use the exponential mixing results for diagonalizable flows on $X$ to give upper estimates for the Hausdorff dimension of the set of points whose trajectories miss $U$. This extends a recent result of Kadyrov and produces new applications to Diophantine approximation, such as an upper bound for the Hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms, generalizing an estimate due to Broderick and Kleinbock.
We prove a criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups. This general observation is then applied to homogeneous spaces and the space of translation surfaces.
We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus. Using the new invariants we significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank $S$-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adelic torus and are closely related to class groups of number fields. The distribut…
The purpose of this article is to describe and characterize the limit distributions of translates of a bounded open "piece of orbit" of a reductive subgroup on a space of S-arithmetic lattices. This is accomplished under a mild assumption of "analytic stability" on the sequence of translates. It is important to note however that it is not necessary to assume that the reductive subgroup or its centralizer are algebraic. Moreover, it is also not necessary to assume that the initial orbit is of finite measure or even closed. This article thus provides important generalizations on previously known…
We prove that infinite orbits of Zariski dense hyperbolic groups equidistribute in homogeneous spaces, in the sense that the family of measures obtained by averaging along spheres in the Cayley graph converges to Haar measure.
Suppose $g_t$ is a $1$-parameter $\mathrm{Ad}$-diagonalizable subgroup of a Lie group $G$ and $Γ< G$ is a lattice. We study the dimension of bounded and divergent orbits of $g_t$ emanating from a class of curves lying on leaves of the unstable foliation of $g_t$ on the homogeneous space $G/Γ$. We obtain sharp upper bounds on the Hausdorff dimension of divergent on average orbits and show that the set of bounded orbits is winning in the sense of Schmidt (and, hence, has full dimension). The class of curves we study is roughly characterized by being tangent to copies of $\mathrm{SL}(2,\mathbb{R}…
