Automorphisms of compact Kähler manifolds with slow dynamics

紧 Kähler 流形自同构的所有阶指数混合与 CLT

Exponential mixing of all orders and CLT for automorphisms of compact K{ä}hler manifolds

本文研究上同调作用简单的紧 Kähler 流形自同构，证明其唯一最大熵测度对 Hölder 可观测量具有所有阶指数混合性，并推出 CLT，结果覆盖所有正熵紧 Kähler 曲面自同构。

一般混合自同构相关性的缓慢衰减

Slow decay of correlations for generic mixing automorphisms

本文证明典型混合自同构的相关性衰减速度不能任意快，并指出轻度混合自同构的非零平均收敛速率不可能达到 $o(1/k_n)$。

The automorphism group of a shift of slow growth is amenable

Suppose $(X,σ)$ is a subshift, $P_X(n)$ is the word complexity function of $X$, and ${\rm Aut}(X)$ is the group of automorphisms of $X$. We show that if $P_X(n)=o(n^2/\log^2 n)$, then ${\rm Aut}(X)$ is amenable (as a countable, discrete group). We further show that if $P_X(n)=o(n^2)$, then ${\rm Aut}(X)$ can never contain a nonabelian free semigroup (and, in particular, can never contain a nonabelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a semigroup.

Slow convergences of ergodic averages

Birkhoff's theorem states that for an ergodic automorphism, the time averages converge to the space average. Given sequence $ψ(n)\to+0$, U. Krengel proved that for any ergodic automorphism there is an indicator such that the corresponding time averages converged a.e. with a rate slower than $ψ$. We prove again similar statements answering a question of I. Podvigin in passing.

慢动力学中的多项式对数体积增长与扭曲齐次坐标环的GK维数

Polynomial log-volume growth in slow dynamics and the GK-dimensions of twisted homogeneous coordinate rings

研究零熵自同构的多项式对数体积增长，推导出新的上下界，并在三维情况下确定其值，同时改进GK维数的界。

非交换代数几何的动力学：在除法代数中膨胀、分支和崩溃的动力学

Dynamical Non-Commutative Algebraic Geometry: Inflation, Bifurcation, and the Dynamics of Collapse across Division Algebras

该研究提出非交换代数几何框架，通过分析多项式根流形的演化和稳定性，揭示了在除法代数中拓扑分支和崩溃的动力学机制，并建立熵缩放定律描述对称性破缺相变。