Dmitry Dolgopyat Giovanni Forni Rostislav Grigorchuk Boris Hasselblatt Anatole Katok Svetlana Katok Dmitry Kleinbock Raphaël Krikorian Jens Marklof
Journal of Modern Dynamics 2008, 2(1): i-v doi: 10.3934/jmd.2008.2.1i
The editors of the Journal of Modern Dynamics are happy to dedicate this issue to Gregory Margulis, who, over the last four decades, has influenced dynamical systems as deeply as few others have, and who has blazed broad trails in the application of dynamical systems to other fields of core mathematics.

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Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.
An application of lattice points counting to shrinking target problems
Dmitry Kleinbock Xi Zhao
Discrete & Continuous Dynamical Systems - A 2018, 38(1): 155-168 doi: 10.3934/dcds.2018007

We apply lattice points counting results to solve a shrinking target problem in the setting of discrete time geodesic flows on hyperbolic manifolds of finite volume.

keywords: Shrinking target problems hyperbolic geometry geodesic flows counting of lattice points
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

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