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JMD

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.

For more information please click the “Full Text” above.

Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.

For more information please click the “Full Text” above.

Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.

keywords:

JMD

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 $\SL_{m+n}(\mathbb R)$/$SL_{m+n}(\mathbb Z)$.

$|Y_i\q - p_i| < \varepsilon e^{-t_i}\,$ $i = 1,\ldots, m$

$|q_j| < \varepsilon e^{t_{m+j}}\,$ $j = 1,\ldots, n$

(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 $\SL_{m+n}(\mathbb R)$/$SL_{m+n}(\mathbb Z)$.

JMD

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.

DCDS

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.

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