Dirichlet's theorem on diophantine approximation and homogeneous flows
Dmitry Kleinbock Barak Weiss
Journal of Modern Dynamics 2008, 2(1): 43-62 doi: 10.3934/jmd.2008.2.43
Given an $m \times n$ real matrix $Y$, an unbounded set $\mathcal{T}$ of parameters $t =( t_1, \ldots, t_{m+n})\in\mathbb{R}_+^{m+n}$ with $\sum_{i = 1}^m t_i =\sum_{j = 1}^{n} t_{m+j} $ and $0<\varepsilon \leq 1$, we say that Dirichlet's Theorem can be $\varepsilon$-improved for $Y$ along $\mathcal{T}$ if for every sufficiently large $\v \in \mathcal{T}$ there are nonzero $\q \in \mathbb Z^n$ and $\p \in \mathbb Z^m$ such that
$|Y_i\q - p_i| < \varepsilon e^{-t_i}\,$     $i = 1,\ldots, m$
$|q_j| < \varepsilon e^{t_{m+j}}\,$     $j = 1,\ldots, n$
(here $Y_1,\ldots,Y_m$ are rows of $Y$). We show that for any $\varepsilon<1$ and any $\mathcal{T}$ 'drifting away from walls', see (1.8), Dirichlet's Theorem cannot be $\epsilon$-improved along $\mathcal{T}$ for Lebesgue almost every $Y$. In the case $m = 1$ we also show that for a large class of measures $\mu$ (introduced in [14]) there is $\varepsilon_0>0$ such that for any drifting away from walls unbounded $\mathcal{T}$, any $\varepsilon<\varepsilon_0$, and for $\mu$-almost every $Y$, Dirichlet's Theorem cannot be $\varepsilon$-improved along $\mathcal{T}$. These measures include natural measures on sufficiently regular smooth manifolds and fractals.
    Our results extend those of several authors beginning with the work of Davenport and Schmidt done in late 1960s. The proofs rely on a translation of the problem into a dynamical one regarding the action of a diagonal semigroup on the space $\SL_{m+n}(\mathbb R)$/$SL_{m+n}(\mathbb Z)$.
keywords: Diophantine approximation equidistribution friendly measures. flows on homogeneous spaces quantitative nondivergence
Modified Schmidt games and a conjecture of Margulis
Dmitry Kleinbock Barak Weiss
Journal of Modern Dynamics 2013, 7(3): 429-460 doi: 10.3934/jmd.2013.7.429
We prove a conjecture of G.A. Margulis on the abundance of certain exceptional orbits of partially hyperbolic flows on homogeneous spaces by utilizing a theory of modified Schmidt games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in mid-1960s.
keywords: Schmidt games homogeneous spaces. conjecture of Margulis partially hyperbolic flows
Dynamics on the infinite staircase
W. Patrick Hooper Pascal Hubert Barak Weiss
Discrete & Continuous Dynamical Systems - A 2013, 33(9): 4341-4347 doi: 10.3934/dcds.2013.33.4341
For the 'infinite staircase' square tiled surface we classify the Radon invariant measures for the straight line flow, obtaining an analogue of the celebrated Veech dichotomy for an infinite genus lattice surface. The ergodic Radon measures arise from Lebesgue measure on a one parameter family of deformations of the surface. The staircase is a $\mathbb{Z}$-cover of the torus, reducing the question to the well-studied cylinder map.
keywords: ergodicity Dynamics Maharam measure infinite staircase infinite lattice surface.

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