arXiv subjects
Publications and source records attributed to Ronggang Shi.
This paper is about topological rigidity of diagonal group actions on the homogeneous $\SL_4\big(\F(\!(t^{-1})\!)\big)/\SL_4(\F[t])$ where $\F$ is a finite field of characteristic $3$. We show that there is a non-closed relatively compact orbit of the diagonal group.
The aim of this paper is to study the product of $n$ linear forms over function fields. We calculate the maximum value of the minima of the forms with determinant one when $n$ is small. The value is equal to the natural bound given by algebraic number theory. Our proof is based on a reduction theory of diagonal group orbits on homogeneous spaces. We also show that the forms defined algebraically correspond to periodic orbits with respect to the diagonal group actions.
Let $G$ be a real Lie group, $Λ<G$ a lattice and $H<G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of $H$-expanding measures $μ$ on $H$ and, applying recent work of Eskin-Lindenstrauss, prove that $μ$-stationary probability measures on $G/Λ$ are homogeneous. Transferring a construction by Benoist-Quint and drawing on ideas of Eskin-Mirzakhani-Mohammadi, we construct Lyapunov/Margulis functions to show that $H$-expanding random walks on $G/Λ$ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in $G/Λ$ for $H$-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons-Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a non-conformal and weighted setting.
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.
We study invariant probability measures on the homogeneous space $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ for the action of subgroups of $\mathrm{SL}_n(\mathbb R)$ of the form $SF$ where $F$ is generated by one parameter unipotent groups and $S$ is a one parameter $\mathbb R$-diagonalizable group normalizing $F$. Under the assumption that $S$ contains an element with only one eigenvalue less than one (counted with multiplicity) and others bigger than one we prove that all the $SF$ invariant and ergodic probability measures on $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ are homogeneous.
We show that effective $2\ell$-multiple correlations imply quantitative $\ell$-multiple pointwise ergodic theorems. The result has a wide class of applications which include subgroup actions on homogeneous spaces, ergodic nilmanifold automorphisms, subshifts of finite type and Young towers.
The dynamics of one parameter diagonal group actions on finite volume homogeneous spaces has a partially hyperbolic feature. In this paper we extend the Livšic type result to these possibly noncompact and nonaccessible systems. We also prove a central limit theorem for the Birkhoff averages of points on a horospherical orbit. The Livšic type result allows us to show that the variance of the central limit theorem is nonzero provided that the test function has nonzero mean with respect to an invariant probability measure.
We prove a version of the Khinchine--Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This correspondence together with a multiple ergodic theorem are used to study rational approximations in several congruence classes simultaneously. The result in this part holds in the generality of weighted approximation but is restricted to simple approximation functions.
Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on finite volume homogeneous space $G/Γ$ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/Γ$ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In the paper we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Ruhr, we prove a multiple ergodic theorem with an error rate.
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.
Let $w=(w_1, w_2)$ be a pair of positive real numbers with $w_1+w_2=1$ and $w_1\ge w_2$. We show that the set of $w$-weighted singular vectors in $\mathbb R^2$ has Hausdorff dimension $2- \frac{1}{1+w_1}$. This extends the previous work of Yitwah Cheung on the Hausdorff dimension of the usual (unweighted) singular vectors in $\mathbb R^2$.
In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. We also prove applications of these results to dynamical billiards, mathematical physics and number theory. In the space of affine lattices $ASL_2(\mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum $\mathcal{H}(1,1)$ of translation surfaces. For these curves (and more in general curves which are well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff genericity) we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, which was recently explored by Dragovic and Radnovic, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. This generalizes a phenomenon recently discovered by Fraczek and Schmoll which could so far only be proved for random periodic configurations. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers, which extends previous work by Elkies and McMullen, is also obtained.
Let $Γ$ be a lattice of a semisimple Lie group $L$. Suppose that one parameter Ad-diagonalizable subgroup $\{g_t\}$ of $L$ acts ergodically on $L/Γ$ with respect to the probability Haar measure $μ$. For certain proper subgroup $U$ of the unstable horospherical subgroup of $\{g_t\}$ we show that given $x\in L/Γ$ for almost every $u\in U$ the trajectory $\{g_tux: 0\le t\le T\}$ is uniformly distributed with respect to $μ$ as $T\to \infty$.
In this note we give a detailed proof of certain results on geometry of numbers in the $S$-adic case. These results are well-known to experts, so the aim here is to provide a convenient reference for the people who need to use them.
Consider $G=\SL_{ d }(\mathbb R)$ and $ Γ=\SL_{ d }(\mathbb Z)$. It was recently shown by the second-named author \cite{s} that for some diagonal subgroups $\{g_t\}\subset G$ and unipotent subgroups $U\subset G$, $g_t$-trajectories of almost all points on all $U$-orbits on $G/Γ$ are equidistributed with respect to continuous compactly supported functions $φ$ on $G/Γ$. In this paper we strengthen this result in two directions: by exhibiting an error rate of equidistribution when $φ$ is smooth and compactly supported, and by proving equidistribution with respect to certain unbounded functions, namely Siegel transforms of Riemann integrable functions on $\R^d$. For the first part we use a method based on effective double equidistribution of $g_t$-translates of $U$-orbits, which generalizes the main result of \cite{km12}. The second part is based on Schmidt's results on counting of lattice points. Number-theoretic consequences involving spiraling of lattice approximations, extending recent work of Athreya, Ghosh and Tseng \cite{agt1}, are derived using the equidistribution result.
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 has full Hausdorff dimension. Our result for numbers is the counterpart to a classical result of Kaufmann and gives a comprehensive understanding.
We show that if $\mathcal{L}$ is a line in the plane containing a badly approximable vector, then almost every point in $\mathcal{L}$ does not admit an improvement in Dirichlet's theorem. Our proof relies on a measure classification result for certain measures invariant under a non-abelian two dimensional group on the homogeneous space $\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$. Using the measure classification theorem, we reprove a result of Shah about planar nondegenerate curves (which are not necessarily analytic), and prove analogous results for the framework of Diophantine approximation with weights. We also show that there are line segments in $\mathbb{R}^3$, which do contain badly approximable points, and for which all points do admit an improvement in Dirichlet's theorem.
Let $λ$ be a probability measure on $\mathbb T^{n-1}$ where $n=2$ or 3. Suppose $λ$ is invariant, ergodic and has positive entropy with respect to the linear transformation defined by a hyperbolic matrix. We get a measure $μ$ on $SL_n(\mathbb Z)\backslash SL_n(\mathbb R)$ by putting $λ$ on some unstable horospherical orbit of the right translation of $a_t=\mathrm{diag}(e^t,..., e^t, e^{-(n-1)t})$ $(t>0)$. We prove that if the average of $μ$ with respect to the flow $a_t$ has a limit, then it must be a scalar multiple of the probability Haar measure. As an application we show that if the entropy of $λ$ is large, then Dirichlet's theorem is not improvable $λ$ almost surely.