Maximal Convergence Groups and Rank One Symmetric Spaces

The unique measure of maximal entropy for a compact rank one locally CAT(0) space

Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which has topological entropy equal to the critical exponent of the Poincaré series.

Free abelian group actions on normal projective varieties: sub-maximal dynamical rank case

Let $X$ be a normal projective variety of dimension $n$ and $G$ an abelian group of automorphisms such that all elements of $G\setminus \{\mathrm{id}\}$ are of positive entropy. Dinh and Sibony showed that $G$ is actually free abelian of rank $\le n - 1$. The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\mathrm{rank} G = n - 2$.

椭圆上极大对称周期轨道的 Lazutkin 坐标

Lazutkin coordinates of the maximal symmetric periodic orbits on the ellipse

本文针对椭圆弹子球系统，利用作用–角变量推导极大对称周期轨道 Lazutkin 坐标的二阶近似，为相关谱刚性证明提供关键计算。

Compact Kähler threefolds with the action of an abelian group of maximal rank

In this note, we study the normal compact Kähler (possibly singular) threefold $X$ admitting the action of a free abelian group $G$ of maximal rank, all the non-trivial elements of which are of positive entropy. If such $X$ is further assumed to have only terminal singularities, then we prove that it is either a rationally connected projective threefold or bimeromorphic to a quasi-étale quotient of a complex $3$-torus.

Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy

We determine positive-dimensional G-periodic proper subvarieties of an n-dimensional normal projective variety X under the action of an abelian group G of maximal rank n-1 and of positive entropy. The motivation of the paper is to understand the obstruction for X to be G-equivariant birational to the quotient variety of an abelian variety modulo the action of a finite group.

Compact Kähler manifolds with automorphism groups of maximal rank

For an automorphism group G on an n-dimensional (n > 2) normal projective variety or a compact Kähler manifold X so that G modulo its subgroup N(G) of null entropy elements is an abelian group of maximal rank n-1, we show that N(G) is virtually contained in Aut_0(X), the X is a quotient of a complex torus T and G is mostly descended from the symmetries on the torus T, provided that both X and the pair (X, G) are minimal.