American Institute of Mathematical Sciences

ISSN:
1930-5311

eISSN:
1930-532X

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Journal of Modern Dynamics

January 2008 , Volume 2 , Issue 1

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 the full preface, please click the "full text" button above. Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson, and Anton Zorich.

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2008, 2(1): i-v doi: 10.3934/jmd.2008.2.1i +[Abstract](1273) +[PDF](723.1KB)
Abstract:
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.

Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.
2008, 2(1): 1-5 doi: 10.3934/jmd.2008.2.1 +[Abstract](843) +[PDF](176.5KB)
Abstract:
This volume is devoted to the mathematical works of Gregory Margulis, and their inﬂuence. The editors have asked me to write about aspects of Margulis’ career that might be of general human interest. I shall restrict myself to experiences that I personally have witnessed. Thus, this will be one colleague’s observations of Grisha’s life in the USSR and in the USA.
2008, 2(1): 7-13 doi: 10.3934/jmd.2008.2.7 +[Abstract](802) +[PDF](218.6KB)
Abstract:
When I received the unexpected but attractive suggestion to write something about Gregory (Grisha) Margulis I began to wonder what I could communicate to the reader. I wanted to avoid a stereotypical anniversary article of the kind seen for example in Russian Mathematical Surveys or Physics Uspechi that lists some biographical dates of the honoree as well as his titles and honors and reviews his scientific results. I will instead tell some episodes of my many years of acquaintance with Grisha. Perhaps, it is better to say that these are some observations on our life, based on our lasting friendship.
For the paper, click the "full text" button above.
2008, 2(1): 15-42 doi: 10.3934/jmd.2008.2.15 +[Abstract](1182) +[PDF](306.6KB)
Abstract:
We consider a semi-simple algebraic group $\mathbf G$ defined over a local field of zero characteristic and we denote by $G$ the group of its $k$-rational points. For $\Gamma$ a "large" sub-semigroup of $G$ we define a closed subgroup 〈Spec$\Gamma$〉 associated with $\Gamma$, and we show that 〈Spec$\Gamma$〉 is large in a certain sense. This allows us to study the $\Gamma$-orbit closures for certain $\Gamma$-actions. The analytic structure of closed subgroups of $G$, over $\mathbb R$ or $\mathbb Q_{p}$, allows to use the Lie algebras techniques. The properties of the limit set of $\Gamma$ are developed ; they play an important role in the proofs.
2008, 2(1): 43-62 doi: 10.3934/jmd.2008.2.43 +[Abstract](1193) +[PDF](275.5KB)
Abstract:
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 $\SL_{m+n}(\mathbb R)$/$SL_{m+n}(\mathbb Z)$.
2008, 2(1): 63-81 doi: 10.3934/jmd.2008.2.63 +[Abstract](1182) +[PDF](223.7KB)
Abstract:
We establish stable ergodicity of diffeomorphisms with partially hyperbolic attractors whose Lyapunov exponents along the central direction are all negative with respect to invariant SRB-measures.
2008, 2(1): 83-128 doi: 10.3934/jmd.2008.2.83 +[Abstract](1250) +[PDF](523.7KB)
Abstract:
We consider measures on locally homogeneous spaces $\Gamma \backslash G$ which are invariant and have positive entropy with respect to the action of a single diagonalizable element $a \in G$ by translations, and prove a rigidity statement regarding a certain type of measurable factors of this action.
This rigidity theorem, which is a generalized and more conceptual form of the low entropy method of [14,3] is used to classify positive entropy measures invariant under a one parameter group with an additional recurrence condition for $G=G_1 \times G_2$ with $G_1$ a rank one algebraic group. Further applications of this rigidity statement will appear in forthcoming papers.
2008, 2(1): 129-138 doi: 10.3934/jmd.2008.2.129 +[Abstract](1134) +[PDF](146.1KB)
Abstract:
Let $Q$ be a nondegenerate indefinite quadratic form on $\mathbb{R}^n$, $n\geq 3$, which is not a scalar multiple of a rational quadratic form, and let $C_Q=\{v\in \mathbb R^n | Q(v)=0\}$. We show that given $v_1\in C_Q$, for almost all $v\in C_Q \setminus \mathbb R v_1$ the following holds: for any $a\in \mathbb R$, any affine plane $P$ parallel to the plane of $v_1$ and $v$, and $\epsilon >0$ there exist primitive integral $n$-tuples $x$ within $\epsilon$ distance of $P$ for which $|Q(x)-a|<\epsilon$. An analogous result is also proved for almost all lines on $C_Q$.
2008, 2(1): 139-185 doi: 10.3934/jmd.2008.2.139 +[Abstract](1065) +[PDF](590.4KB)
Abstract:
Moduli spaces of Abelian and quadratic differentials are stratiﬁed by multiplicities of zeroes; connected components of the strata correspond to ergodic components of the Teichmüller geodesic ﬂow. It is known that the strata are not necessarily connected; the connected components were recently classiﬁed by M. Kontsevich and the author and by E. Lanneau. The strata can be also viewed as families of ﬂat metrics with conical singularities and with $\mathbb Z$/$2 \mathbb Z$-holonomy.
For every connected component of each stratum of Abelian and quadratic differentials we construct an explicit representative which is a Jenkins–Strebel differential with a single cylinder. By an elementary variation of this construction we represent almost every Abelian (quadratic) differential in the corresponding connected component of the stratum as a polygon with identiﬁed pairs of edges, where combinatorics of identiﬁcations is explicitly described.
Speciﬁcally, the combinatorics is expressed in terms of a generalized permutation. For any component of any stratum of Abelian and quadratic differentials we construct a generalized permutation in the corresponding extended Rauzy class.

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