环面上的仿射随机游走

Tame semicascades and cascades generated by affine self-mappings of the $d$-torus

We give a complete characterization of the affine self-mappings $φ$ of the torus $\mathbb T^d$ that generate Köhler-tame semicascades and cascades. Namely, we show that the semicascade generated by $φ$ is tame if and only if the matrix $A$ of $φ$ satisfies $A^p=A^q$, where $p$ and $q$ are some nonnegative integers, $p\neq q$. For cascades the corresponding condition has the form $A^m=I$, where $m$ is some positive integer and $I$ is the identity matrix.

Discrepancy Skew Products and Affine Random Walks

We prove bounded rational ergodicity for some discrepancy skew products whose rotation number has bad rational approximation. This is done by considering the asymptotics of associated affine random walks.

Extreme value theory for random walks on homogeneous spaces

In this paper we study extreme events for random walks on homogeneous spaces. We consider the following three cases. On the torus we study closest returns of a random walk to a fixed point in the space. For a random walk on the space of unimod- ular lattices we study extreme values for lengths of the shortest vector in a lattice. For a random walk on a homogeneous space we study the maximal distance a random walk gets away from an arbitrary fixed point in the space. We prove an exact limiting distribution on the torus and upper and lower bounds for sparse subsequences of random walks in the tw…

Decomposition of random walk measures on the one-dimensional torus

The main result of this paper is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset $S$ of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one $μ_1$ has the property that the random walk with initial distribution $μ_1$ evolved by the action of $S$ equidistributes very fast. The second measure $μ_2$ in the decomposition is concentrated on very small neighborhoods of a small number of points.

Equation cohomologique d'un automorphisme affine hyperbolique du tore

We study the discrete cohomological equation of a hyperbolic affine automorphism Y of the torus (whose linear part is not necessarily diagonalisable). More precisely ; if d is the cobord operator defined by : d (h) = h-hoY for every element h of the Fréchet space E of the differentiable functions on the torus, we show that the image Im(d) is a closed of E and that consequently the space of cohomology H^1(Y; E):=E/Im(d) is a nontrivial Fréchet space. We also prove the existence of a continuous linear operator L defined from Im(d) to E such that for every element g of Im(d), the image f = L(g) i…

KAM-rigidity for parabolic affine abelian actions

We show the following dichotomy for a linear parabolic $\mathbb Z^2$-action $ρ_L$ on the torus with at least one step-2 generator: (i) Any affine $\mathbb Z^2$-action with linear part $ρ_L$ has a $\mathbb Z$-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine $\mathbb Z^2$-action with linear part $ρ_L$ is KAM-rigid under volume preserving perturbations.

关于模空间上随机游走的收敛性

On Convergence of Random Walks on Moduli Space

本文证明了阿贝尔微分模空间上随机游走在两种模式下的收敛性，包括n步分布收敛和几乎必然路径等分布，推广了齐次空间上的已有结果。

Quantisations of piecewise affine maps on the torus and their quantum limits

For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions. We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we ca…

Rigidity of measurable structure for Z^d-actions by automorphisms of a torus

We show that for certain classes of actions of Z^d, d >= 2, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we consruct various examples of Z^d-actions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomor…

On the affine random walk on the torus

Let $μ$ be a borelian probability measure on $\mathbf{G}:=\mathrm{SL}_d(\mathbb{Z}) \ltimes \mathbb{T}^d$. Define, for $x\in \mathbb{T}^d$, a random walk starting at $x$ denoting for $n\in \mathbb{N}$, \[ \left\{\begin{array}{rcl} X_0 &=&x\\ X_{n+1} &=& a_{n+1} X_n + b_{n+1} \end{array}\right. \] where $((a_n,b_n))\in \mathbf{G}^\mathbb{N}$ is an iid sequence of law $μ$. Then, we denote by $\mathbb{P}_x$ the measure on $(\mathbb{T}^d)^\mathbb{N}$ that is the image of $μ^{\otimes \mathbb{N}}$ by the map $\left((g_n) \mapsto (x,g_1 x, g_2 g_1 x, \dots , g_n \dots g_1 x, \dots)\right)$ and for an…

仿射群的随机自由半群

Random free semigroups of affine groups

研究可解非幂零情形下半群的随机自由性，聚焦相关仿射模型，表明自由性长期行为由阿贝尔影子决定，证明结合仿射随机游走局部收缩定理与乒乓论证。