Non-homogeneous locally free actions of the affine group

关于齐次自仿测度的Rajchman性质

On the Rajchman property of homogeneous self-affine measures

本文完整刻画了傅里叶变换不趋于零的齐次自仿测度，证明整数自仿测度具有幂次傅里叶衰减等价于绝对连续，并给出判定绝对连续概率向量的算法。

Fourier decay for homogeneous self-affine measures

We show that for Lebesgue almost all $d$-tuples $(θ_1,\ldots,θ_d)$, with $|θ_j|>1$, any self-affine measure for a homogeneous non-degenerate iterated function system $\{Ax+a_j\}_{j=1}^m$ in ${\mathbb R}^d$, where $A^{-1}$ is a diagonal matrix with the entries $(θ_1,\ldots,θ_d)$, has power Fourier decay at infinity.

KAM rigidity for partially hyperbolic affine Z^k actions on the torus with a rank one factor

We show that ergodic affine abelian discrete actions on the torus, that have a rank-one factor in their linear part, are locally rigid in a KAM sense if and only if the rank one factor is trivial and the action is higher-rank transversally to this factor. Since it has been proved by Damjanovic and Katok that affine actions with higher-rank linear part are locally rigid, our result completes the local rigidity picture for affine actions on the torus.

Rigidity for group actions on homogeneous spaces by affine transformations

We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $Λ$ a lattice in $G$, and $Γ$ a subgroup of the affine group Aff$(G)$ stabilizing $Λ$. Then the action of $Γ$ on $G/Λ$ has the rigidity property in the sense of S. Popa, if and only if the induced action of $Γ$ on $\mathbb{P}(\frak{g})$ admits no $Γ$-invariant probability measure, where $\frak{g}$ is the Lie algebra of $G$. This generalizes results of M. Burger, and A. Ioana and Y. Shalom. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nil…

Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows

In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral …

Locally unipotent invariant measures and limit distribution of a sequence of polynomial trajectories on homogeneous spaces

Let $G$ be a Lie group and $Γ$ be a lattice in $G$. We introduce the notion of locally unipotent invariant measures on $G/Γ$. We then prove that under some conditions, the limit measure supported on the image of polynomial trajectories on $G/Γ$ is locally unipotent invariant, thus give a partial answer to an equidistribution problem for higher dimensional polynomial trajectories on homogeneous spaces, which was raised by Shah in \cite{shah1994limit}. The proof relies on Ratner's measure classification theorem, linearization technique for polynomial trajectories near singular sets and a twistin…