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A number is in BA if its orbit is bounded. Any such orbit closure must contain a full $A=\langle g_t\rangle$ orbit. By examining the possible subgroups, any such hypothetical $H$, as a stability group …

Okay, these are just basic technical tidbits. Unsure if anything here is of research level. $SO(n)$ acts transitively on the sphere of $\mathbb{R}^{n}$ (there's a slight ambiguity about $0$, but I gue …

This thing cannot hold no matter what. As Echo rightfully commented, the expression doesn't even compile when $t=0$. It is true one might temper the integral away from $0$, but that's not what you ask …

Notice that every $m\in SL_{n}(\mathbb{R})$ can be written as $m=u\cdot q$, where $q$ is an ``opposite horospherical'' (maybe with center), namely it is not getting expanded by the $g_{t}$ action. The …

First of all, $Y$ is not called the “grid space”. It is sometimes called the affine space and can be identified with a quotient of the affine group $\operatorname{ASL}_{n}$, namely the semi-direct pro …

The connections between Dynamics and Lie Groups (or Algebraic groups) comes mainly in two flavours: Smooth dynamics, like others have stated Hamiltonian dyanmics and differential equations. Applicati …

Ok, let take $v$ in the bundle at $x$. We may decompose $v=v^{u}+v^{s}\in E^{u}\oplus E^{s}$. Assume without loss of generality that $\lVert v\rVert=1$. Applying $A^{i}$, using equivariance and Osceld …

It is evident that the singular vectors are defined as the ``$u_{A}$-part which is $g_{t}$ divergent in the future'', this gives $m\cdot n$ ($=\dim \left(u_{A}\right)$) minus the dimension of the sing …

$\DeclareMathOperator\supp{supp}$The theorem holds for semigroups as well (well, in the finite volume setting! in the infinite volume setting there are subtleties between two-sided and one-sided avera …

Show that for a Bernoulli system, there exists ergodic (Bernoulli) measures of any given entropy (between 0 and full entropy). Pick such a measure with appropriate entropy as you would like. Recall t …

There's a nice proof by Margulis showing that arithmetic subgroups are indeed lattices using the famous Dani-Margulis non-divergence theorem. Actually if you will investigate Ratner's original formula …

While Peter Humphries' answer is entirely correct for the question asked by the OP, the technique indicated there is far from addressing the most general situation. The most basic technique towards t …

EDIT - The answer below deals with an ergodic m.p.s As this question got up-voted, I've decided to fuly write a solution, based on the sketch I've made in the comments. Fix some $\varepsilon>0$ smal …

This is indeed true for some "nice systems", for example one can show this theorem (for say $L^{2}$-functions) for Kronecker systems simply by van-der-Corput trick. In general, those averages converg …

The first question is false as stated. By Artin's encoding, geodesics on $SL_{2}(\mathbb{R})/SL_{2}(\mathbb{Z})$ corresponding to continued fractions, and the geodesic flow corresponds to the shift. I …