In this paper we compute the Hausdorff dimension of the set D_n of points on divergent trajectories of the homogeneous flow induced by a certain one-parameter subgroup of G=SL(2,R) acting by left multiplication on the product space G^n/Gamma^n, where Gamma=SL(2,Z). We prove that the Hausdorff dimension of D_n equals 3n-(1/2) for any n greater than one.
For one parameter subgroup action on a finite volume homogeneous space, we consider the set of points admitting divergent on average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Y. Cheung.
In this article, we consider the product space of several non-compact finite volume hyperbolic spaces, $V_1, V_2, \dots , V_k$ of dimension $n$. Let $\mathrm{T}^1(V_i)$ denote the unit tangent bundle of $V_i$ for each $i=1,\dots , k$, then for every $(v_1, \dots , v_k) \in \mathrm{T}^1 (V_1) \times \cdots \times \mathrm{T}^1 (V_k)$, the diagonal geodesic flow $g_t$ is defined by $g_t (v_1, \dots , v_k) = (g_t v_1, \dots , g_t v_k)$. And we define $$\mathfrak{D}_k =\left\{ (v_1, \dots, v_k) \in \mathrm{T}^1 (V_1) \times \cdots \times \mathrm{T}^1 (V_k): g_t(v_1, \dots, v_k) \text{ divergent, as…
In this paper we show that the Hausdorff dimension of the set of singular pairs is 4/3. We also show that the action of diag(e^t,e^t,e^{-2t}) on SL(3,R)/SL(3,Z) admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates, answering a question of A.N. Starkov. As a by-product of the analysis, we obtain a higher dimensional generalisation of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the techniques used to compute Hausdorff dimension, we show that the set of real numbers with divergent partial quotients has Hausdorff…
The set of directions from a quadratic differential that diverge on average under Teichmuller geodesic flow has Hausdorff dimension exactly equal to one-half.
The periodic wind-tree model is a family T(a,b) of billiards in the plane in which identical rectangular scatterers of size axb are disposed at each integer point. It was proven by P. Hubert, S. Lelièvre and S. Troubetzkoy (arXiv:0912.2891v1) that for a residual set of parameters (a,b) the billiard flow in T(a,b) is recurrent in almost every direction. We prove that for many parameters (a,b) there exists a set S of angles of positive Hausdorff dimension such that every billiard trajectory in T(a,b) with initial angle in S is self-avoiding. In particular, the flow in a direction of S is diverge…
We prove a modified version for a conjecture of Weiss from 2004. Let $G$ be a semisimple real algebraic group defined over $\mathbb{Q}$, $Γ$ be an arithmetic subgroup of $G$. A trajectory in $G/Γ$ is divergent if eventually it leaves every compact subset, and is obvious divergent if there is a finite collection of algebraic data which cause the divergence. Let $A$ be a diagonalizable subgroup of $G$ of positive dimension. We show that if the projection of $A$ to any $\mathbb{Q}$-factor of $G$ is of small enough dimension (relatively to the $\mathbb{Q}$-rank of the $\mathbb{Q}$-factor), then th…
Let $G$ be a real algebraic group defined over $\mathbb{Q}$, $Γ$ be an arithmetic subgroup of $G$, and $T$ be a maximal $\mathbb{R}$-split torus. A trajectory in $G/Γ$ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which account for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in $T$, we study the existence of non-obvious divergent trajectories under its action in $G/Γ$. We get a sufficient condition for the existence of a non-obvious divergence trajectory i…
In this paper, we determine the Hausdorff dimension of the set of points with divergent trajectories on the product of certain homogeneous spaces. The flow is allowed to be weighted with respect to the factors in the product space. The result is derived from its counterpart in Diophantine approximation. In doing this, we introduce a notion of jointly singular matrix tuples, and extend the dimension formula for singular matrices to such matrix tuples.
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}…
Let $H = \mathrm{SO}(n,1)$ and $A = \{a(t): t \in \mathbb{R}\}$ be a maximal $\mathbb{R}$-split Cartan subgroup of $H$. Let $Γ\subset H \times H$ be a nonuniform lattice in $H \times H$ and $X_Γ : = H \times H/ Γ$. Let $A_2 : = \{ a_2(t):=a(t) \times a(t) : t \in \mathbb{R}\} \subset A\times A$ on $X_Γ$ and $\mathcal{D}_Γ\subset X_Γ$ denote the collection of points $x \in X_Γ$ such that $a_2(t)x$ diverges as $t \rightarrow +\infty$. In this note, we will show that if the Hausdorff dimension of $\mathcal{D}_Γ$ is greater than $\dim (H\times H) - 2(n-1)$, then $Γ$ is essentially split, namely, i…
We describe the multifractal nature of random weak Gibbs measures on some class of attractors associated with $C^1$ random dynamics semi-conjugate to a random subshift of finite type. This includes the validity of the multifractal formalism, the calculation of Hausdorff and packing dimensions of the so-called level sets of divergent points, and a $0$-$\infty$ law for the Hausdorff and packing measures of the level sets of the local dimension.
In this paper we study the dimension of a family of sets arising in open dynamics. We use exponential mixing results for diagonalizable flows in compact homogeneous spaces $X$ to show that the Hausdorff dimension of set of points that lie on trajectories missing a particular open ball of radius $r$ is at most $$\dim X + C\frac{r^{\dim X}}{\log r},$$ where $C>0$ is a constant independent of $r>0$. Meanwhile, we also describe a general method for computing the least cardinality of open covers of dynamical sets using volume estimates.
For any $β> 1$, denoted by $r_n(x,β)$ the maximal length of consecutive zeros amongst the first $n$ digits of the $β$-expansion of $x\in[0,1]$. The limit superior (respectively limit inferior) of $\frac{r_n(x,β)}{n}$ is linked to the classical Diophantine approximation (respectively uniform Diophantine approximation). We obtain the Hausdorff dimension of the level set $$E_{a,b}=\left\{x \in [0,1]: \liminf_{n\rightarrow \infty}\frac{r_n(x,β)}{n}=a,\ \limsup_{n\rightarrow \infty}\frac{r_n(x,β)}{n}=b\right\}\ (0\leq a\leq b\leq1).$$ Furthermore, we show that the extremely divergent set $E_{0,1}$ …
Singular points for cone actions on the product of certain homogeneous spaces
Institute for Theoretical Sciences, School of Science, Westlake University（西湖大学理学院理论科学研究院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one- parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of singular systems of linear forms (equivalently the set of points with divergent trajectories) as well as the dimension of the set of points with trajectories 'escaping on average' (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheun…
We extend results of parametric geometry of numbers to a general diagonal flow on the space of lattices. Moreover, we compute the Hausdorff dimension of the set of trajectories with every given behavior, with respect to a nonstandard metric and thereby attain bounds on the standard ones.
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.
Let $Δ\subsetneq \mathbb C$ be a simply connected domain, let $f:\mathbb D \to Δ$ be a Riemann map and let $\{z_k\}\subset Δ$ be a compactly divergent sequence. Using Gromov's hyperbolicity theory, we show that $\{f^{-1}(z_k)\}$ converges non-tangentially to a point of $\partial \mathbb D$ if and only if there exists a simply connected domain $U\subsetneq \mathbb C$ such that $Δ\subset U$ and $Δ$ contains a tubular hyperbolic neighborhood of a geodesic of $U$ and $\{z_k\}$ is eventually contained in a smaller tubular hyperbolic neighborhood of the same geodesic. As a consequence we show that i…
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.
