Diagonal actions in positive characteristic

We prove positive characteristic analogues of certain measure rigidity theorems in characteristic zero. More specifically we give a classification result for positive entropy measures on quotients of $\operatorname{SL}_d$ and a classification of joinings for higher rank actions on simply connected absolutely almost simple groups.Comment: 44 page

Spectral gap in the group of affine transformations over prime fields

We study random walks on the groups $\Bbb F^d_p \rtimes$ SL$_d$($\Bbb F_p$). We estimate the spectral gap in terms of the spectral gap of the projection to the linear part SL$_d$($\Bbb F_p$). This problem is motivated by an analogue in the group $\Bbb R^d \rtimes$ SO($d$), which have application to smoothness of self-similar measures.European Research Council (Advanced Research Grant ID: 267259), ISF (grant ID: 983/09), Simons Foundatio

Singular systems of linear forms and non-escape of mass in the space of lattices

Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of oneparameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdor dimension of the set of singular systems of linear forms (equivalently the set of lattices with divergent trajectories) as well as the dimension of the set of lattices with trajectories `escaping on average' (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung. Our method di ers considerably from that of Cheung and Chevallier, and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes

Singular systems of linear forms and non-escape of mass in the space of lattices

Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of oneparameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdor dimension of the set of singular systems of linear forms (equivalently the set of lattices with divergent trajectories) as well as the dimension of the set of lattices with trajectories `escaping on average' (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung. Our method di ers considerably from that of Cheung and Chevallier, and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes

Amenability of algebras of approximable operators

We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson's space.Comment: 20 pages, to appear in Israel Journal of Mathematic

Selective amplification of scars in a chaotic optical fiber

In this letter we propose an original mechanism to select scar modes through coherent gain amplification in a multimode D-shaped fiber. More precisely, we numerically demonstrate how scar modes can be amplified by positioning a gain region in the vicinity of specific points of a short periodic orbit known to give rise to scar modes

Return times, recurrence densities and entropy for actions of some discrete amenable groups

Results of Wyner and Ziv and of Ornstein and Weiss show that if one observes the first k outputs of a finite-valued ergodic process, then the waiting time until this block appears again is almost surely asymptotic to $2^{hk}$, where $h$ is the entropy of the process. We examine this phenomenon when the allowed return times are restricted to some subset of times, and generalize the results to processes parameterized by other discrete amenable groups. We also obtain a uniform density version of the waiting time results: For a process on $s$ symbols, within a given realization, the density of the initial $k$-block within larger $n$-blocks approaches $2^{-hk}$, uniformly in $n>s^k$, as $k$ tends to infinity. Again, similar results hold for processes with other indexing groups.Comment: To appear in Journal d'Analyse Mathematiqu

Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms

We exhibit scarring for certain nonlinear ergodic toral automorphisms. There are perturbed quantized hyperbolic toral automorphisms preserving certain co-isotropic submanifolds. The classical dynamics is ergodic, hence in the semiclassical limit almost all eigenstates converge to the volume measure of the torus. Nevertheless, we show that for each of the invariant submanifolds, there are also eigenstates which localize and converge to the volume measure of the corresponding submanifold.Comment: 17 page