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Journal of Modern Dynamics

2009 , Volume 3 , Issue 3

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Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups
Felipe A. Ramírez
2009, 3(3): 335-357 doi: 10.3934/jmd.2009.3.335 +[Abstract](165) +[PDF](243.0KB)
Abstract:
We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra - one nilpotent and the other semisimple. Second, we consider actions arising from two commuting unipotent flows that come from an embedded copy of $\overline{\SL(2,\RR)}^{k} \times \overline{\SL(2,\RR)}^{l}$. In both cases we show that any smooth $\RR$-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function. These results build on theorems of D. Mieczkowski, where the same is shown for actions on $(\SL(2,\RR) \times \SL(2,\RR))$/Γ.
Logarithm laws for unipotent flows, I
Jayadev S. Athreya and Gregory A. Margulis
2009, 3(3): 359-378 doi: 10.3934/jmd.2009.3.359 +[Abstract](264) +[PDF](200.2KB)
Abstract:
We prove analogs of the logarithm laws of Sullivan and Kleinbock--Margulis in the context of unipotent flows. In particular, we obtain results for one-parameter actions on the space of lattices SL(n, $\R$)/SL(n, $\Z$). The key lemma for our results says the measure of the set of unimodular lattices in $\R^n$ that does not intersect a 'large' volume subset of $\R^n$ is 'small'. This can be considered as a 'random' analog of the classical Minkowski Theorem in the geometry of numbers.
Discontinuity-growth of interval-exchange maps
Christopher F. Novak
2009, 3(3): 379-405 doi: 10.3934/jmd.2009.3.379 +[Abstract](215) +[PDF](347.3KB)
Abstract:
For an interval-exchange map $f$, the number of discontinuities $d(f^n)$ either exhibits linear growth or is bounded independently of $n$. This dichotomy is used to prove that the group $\mathcal{E}$ of interval-exchanges does not contain distortion elements, giving examples of groups that do not act faithfully via interval-exchanges. As a further application of this dichotomy, a classification of centralizers in $\mathcal{E}$ is given. This classification is used to show that $\text{Aut}(\mathcal{E}) \cong \mathcal{E}$ $\mathbb{Z}$/$ 2 \mathbb{Z}$.
Floer homology for negative line bundles and Reeb chords in prequantization spaces
Peter Albers and Urs Frauenfelder
2009, 3(3): 407-456 doi: 10.3934/jmd.2009.3.407 +[Abstract](189) +[PDF](473.0KB)
Abstract:
We prove existence of Reeb orbits for Bohr - Sommerfeld Legendrians in certain prequantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fiberwise quadratic Hamiltonian functions on negative line bundles.
On a generalization of Littlewood's conjecture
Uri Shapira
2009, 3(3): 457-477 doi: 10.3934/jmd.2009.3.457 +[Abstract](211) +[PDF](243.0KB)
Abstract:
We present a class of lattices in $\R^d$ ($d\ge 2$) which we call grid-Littlewood lattices and conjecture that any lattice is such. This conjecture is referred to as GLC. Littlewood's conjecture amounts to saying that $\Z^2$ is grid-Littlewood. We then prove the existence of grid-Littlewood lattices by first establishing a dimension bound for the set of possible exceptions. The existence of vectors (grid-Littlewood-vectors) in $\R^d$ with special Diophantine properties is proved by similar methods. Applications to Diophantine approximations are given. For dimension $d\ge 3$, we give explicit constructions of grid-Littlewood lattices (and in fact lattices satisfying a much stronger property). We also show that GLC is implied by a conjecture of G. A. Margulis concerning bounded orbits of the diagonal group. The unifying theme of the methods is to exploit rigidity results in dynamics ([4, 1, 5]), and derive results in Diophantine approximations or the geometry of numbers.

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