半单李群商空间上单参数幂零流的时变刚性

Parabolic perturbations of unipotent flows on compact quotients of $\text{SL}(3,\mathbb{R})$

We consider a family of smooth perturbations of unipotent flows on compact quotients of $\text{SL}(3,\mathbb{R})$ which are not time-changes. More precisely, given a unipotent vector field, we perturb it by adding a non-constant component in a commuting direction. We prove that, if the resulting flow preserves a measure equivalent to Haar, then it is parabolic and mixing. The proof is based on a geometric shearing mechanism together with a non-homogeneous version of Mautner Phenomenon for homogeneous flows. Moreover, we characterize smoothly trivial perturbations and we relate the existence of…

Polynomial mixing for time-changes of unipotent flows

Let $G$ be a connected semisimple Lie group with finite centre, and let $M= Γ\backslash G$ be a compact homogeneous manifold. Under a spectral gap assumption, we show that smooth time-changes of any unipotent flow on $M$ have polynomial decay of correlations. Our result applies also in the case where $M$ is a finite volume, non-compact quotient under some additional assumptions on the generator of the time-change. This generalizes a result by Forni and Ulcigrai (JMD, 2012) for smooth time-changes of horocycle flows on compact surfaces.

Measure rigidity for solvable group actions in the space of lattices

We study invariant probability measures on the homogeneous space $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ for the action of subgroups of $\mathrm{SL}_n(\mathbb R)$ of the form $SF$ where $F$ is generated by one parameter unipotent groups and $S$ is a one parameter $\mathbb R$-diagonalizable group normalizing $F$. Under the assumption that $S$ contains an element with only one eigenvalue less than one (counted with multiplicity) and others bigger than one we prove that all the $SF$ invariant and ergodic probability measures on $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ are h…

Rigidity of joinings for some measure preserving systems

We introduce two properties: strong R-property and $C(q)$-property, describing a special way of divergence of nearby trajectories for an abstract measure preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$-property. Moreover, we show that if $u_t$ is a unipotent flow on $G/Γ$ with $Γ$ irreducible, then $u_t$ satisfies the $C(q)$-property provided that $u_t$ is not of the form $h_t\times\operatorname{id}$, where $h_t$ is the classical horocycle flow. Finally, we show that the strong R-property hol…

New time-changes of unipotent flows on quotients of Lorentz groups

We study the cocompact lattices $Γ\subset SO(n,1)$ so that the Laplace-Beltrami operator $Δ$ on $SO(n)\backslash SO(n,1)/Γ$ has eigenvalues in $(0,1/4)$, and then show that there exist time-changes of unipotent flows on $SO(n,1)/Γ$ that are not measurably conjugate to the unperturbed ones. A main ingredient of the proof is a stronger version of the branching of the complementary series. Combining it with a refinement of the works of Ratner and Flaminio-Forni is adequate for our purpose.

Anosov流乘积的时间变换的刚性现象

Rigidity phenomena for time changes of products of Anosov flows

研究Anosov流乘积的时间变换的刚性，证明具有相同周期轨道稳定子的时间变换共轭于自同构，以及若周期轨道稳定子可同时对角化则时间变换共轭于流乘积，并应用于3维流形上的C^1时间变换。

不稳定性叶的刚性

Rigidity of the unstable foliation

本文研究了transitive Anosov流在3流形上不稳定性叶的刚性问题，证明了不稳定性叶等价性下流的拓扑共轭性，扩展了Ratner关于负曲率表面水平流的刚性定理。

Smooth rigidity of uniformly quasiconformal Anosov flows

We classify quasiconformal Anosov flows whose strong stable and unstable distributions are at least two dimensional and the sum of these two distributions is smooth. We deduce from this classification result the complete classification of volume-preserving quasiconformal diffeomorphisms whose stable and unstable distributions are at least two dimensional. Our central idea is to take a good time change so that perodic orbits are equi-distributed with respect to a lebesgue measure.

Rigidity of a class of smooth singular flows on $\mathbb T^2$

We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field $\mathcal{X}$ on $\mathbb T^2\setminus \{a\}$, where $\mathcal{X}$ is not defined at $a\in \mathbb T^2$. It follows that the phase space can be decomposed into a (topological disc) $D_\mathcal{X}$ and an ergodic component $E_\mathcal{X}=\mathbb T^2\setminus D_\mathcal{X}$. Let $ω_\mathcal{X}$ be the 1-form associated to $\mathcal{X}$. We show that if $|\int_{E_{\mathcal{X}_1}}dω_{\mathcal{X}_1}|\neq |\int_{E_{\mathcal{X}_2}}dω_{\mathcal{X}_2}|$, then the corresponding flows…