Dynamics on an infinite surface with the lattice property

Haagerup property and Kazhdan pairs via ergodic infinite measure preserving actions

It is shown that a locally compact second countable group $G$ has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free $G$-action $T=(T_g)_{g\in G}$ on an infinite $σ$-finite standard measure space $(X,μ)$ admitting an exhausting $T$-Følner sequence (i.e. a sequence $(A_n)_{n=1}^\infty$ of measured subsets of finite measure such that $A_1\subset A_2\subset\cdots$, $\bigcup_{n=1}^\infty A_n=X$ and $\lim_{n\to\infty}\sup_{g\in K}\frac{μ(T_gA_n\triangle A_n)}{μ(A_n)}= 0$ for each compact $K\subset G$). It is also shown that a pair of groups $H\sub…

Normal forms and Sternberg conjugation theorems for infinite dimensional coupled map lattices

In this paper we present local Sternberg conjugation theorems near attracting fixed points for lattice systems. The interactions are spatially decaying and are not restricted to finite distance. The conjugations obtained retain the same spatial decay. In the presence of resonances the conjugations are to a polynomial normal form that also has decaying properties.

Strong amenability and the infinite conjugacy class property

A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable discrete group is strongly amenable if and only if none of its quotients have the infinite conjugacy class property.

Embedding infinite cyclic covers of knot spaces into 3-space

We say a knot $k$ in the 3-sphere $\mathbb S^3$ has {\it Property $IE$} if the infinite cyclic cover of the knot exterior embeds into $\mathbb S^3$. Clearly all fibred knots have Property $IE$. There are infinitely many non-fibred knots with Property $IE$ and infinitely many non-fibred knots without property $IE$. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property $IE$, then its Alexander polynomial $Δ_k(t)$ must be either 1 or $2t^2-5t+2$, and we give two infinite families of non-fibred genus 1 knots with Property $IE$…

无限熵随机游走的 Liouville 性质与 Poisson 边界：哪里出了问题？

Liouville property and Poisson boundary of random walks with infinite entropy: what's amiss?

本文探讨群上随机游走在步分布熵无限情形下出现的新性质，聚焦 Liouville 性质与 Poisson 边界相关问题。

Groups with infinite FC-center have the Schmidt property

We show that every countable group with infinite FC-center has the Schmidt property, i.e., admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As its consequence, every countable, inner amenable group with property (T) has the Schmidt property.

Extremal Property of the Square Lattice

Motivated by a 2019 result of Faulhuber-Steinerberger, we demonstrate that the real square lattice $\mathbb{Z}^2$ exhibits the same local, extremal property as the hexagonal lattice $Λ$, where distances of lattice points from the `deep holes' of natural fundamental domains increase under perturbation. If $Δ$ is a small perturbation of $\mathbb{Z}^2$ in the space of unimodular lattices, consider $C_r$, the set of points in $A_r$ shifted to $Δ$. If $Δ$ is a perturbation of the lattice $\mathbb{Z}^2$ with respect to the Euclidean metric, then for a fixed deep hole $p$, the summed total distance o…

Choquet-Deny groups and the infinite conjugacy class property

A countable discrete group $G$ is called Choquet-Deny if for every non-degenerate probability measure $μ$ on $G$ it holds that all bounded $μ$-harmonic functions are constant. We show that a finitely generated group $G$ is Choquet-Deny if and only if it is virtually nilpotent. For general countable discrete groups, we show that $G$ is Choquet-Deny if and only if none of its quotients has the infinite conjugacy class property. Moreover, when $G$ is not Choquet-Deny, then this is witnessed by a symmetric, finite entropy, non-degenerate measure.

Rational weak mixing in infinite measure spaces

Rational weak mixing is a measure theoretic version of Krickeberg's strong ratio mixing property for infinite measure preserving transformations. It requires "{\tt density}" ratio convergence for every pair of measurable sets in a dense hereditary ring. Rational weak mixing implies weak rational ergodicity and (spectral) weak mixing. It is enjoyed for example by Markov shifts with Orey's strong ratio limit property. The power, subsequence version of the property is generic.

Arnold diffusion in Hamiltonian systems on infinite lattices

We consider a system of infinitely many penduli on an $m$-dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next-to-nearest neighbor or long range as long as it is strongly decaying. The transfer of energy is given by an Arnold diffusion mechanism which relies on the original V. I Arnold approach: to construct a sequence of hyperbolic invariant quasiperiodic tori with …

Statistical properties of coupled expanding maps on a lattice with general infinite range couplings and Hölder densities

We continue the development of transfer operator techniques for expanding maps on a lattice coupled by general interaction functions. We obtain a spectral gap for an appropriately defined transfer operator, and, as corollaries, the existence of an invariant conformal probability measure for the system, exponential decay of correlations, the central limit theorem and the almost sure invariance principle.

无限晶格系统的耗散动力学

Dissipative dynamics for infinite lattice systems

本文利用非交换狄利克雷型和CCR代数构造无限晶格多粒子系统的耗散动力学，通过态的拟不变性构造酉表示，并证明具有局部守恒量的模型无谱隙且以多项式速率趋于平衡。