Rigidity of measures invariant under the action of a multiplicative semigroup of polynomial growth on $\T$

Hausdorff dimension for fractals invariant under the multiplicative integers

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$ for all $k$. We compute the Hausdorff and Minkowski dimensions of these sets and show that they are typically different. The proof proceeds via a variational principle for multiplicative subshifts.

刚性、扩张与熵在非阿基米德动力学中

Rigidité, expansion et entropie en dynamique non-archimédienne (Rigidity, expansion and entropy in non-Archimedean dynamics)

该研究在非阿基米德动力学中证明了有理映射的刚性性质，表明其拓扑熵为整数对数，并建立了李亚普诺夫指数与野分支位置的关系。

The 3x+1 Semigroup

The 3x+1 semigroup is the multiplicative semigroup generated by the rational numbers of form (2k+1)/(3k+2) for non-negative k, together with 2. This semigroup encodes backward iteration under the 3x+1 map, and the 3x+1 conjecture implies that it contains every positive integer. We prove this is the case, and show that this semigroup consists of all positive rational numbers a/b such that 3 does not divide b.

具乘性噪声的非局部随机Swift-Hohenberg方程随机吸引子的上半连续性及不变测度的存在性

Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise

本文研究了具乘性噪声的2D非局部随机Swift-Hohenberg方程的长时间动力学行为，利用解析半群理论证明了随机吸引子的上半连续性，并推广随机Gronwall引理以建立遍历不变测度的存在性。

Some measure rigidity and equidistribution results for $β$-maps

We prove $\times a$ $\times b$ measure rigidity for multiplicatively independent pairs when $a\in\mathbb{N}$ and $b>1$ is a ``specified'' real number (the $b$-expansion of $1$ has a tail or bounded runs of $0$'s) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along $\times b$ orbits. We also prove a quantitative version of this decay under stronger conditions on the $\times a$ invariant measure. The quantitative version together with the $\times b$ invariance of the limit measure is a step toward a general Host-type po…

Rigidity of contractions on Hilbert spaces

We study the asymptotic behaviour of contractive operators and strongly continuous semigroups on separable Hilbert spaces using the notion of rigidity. In particular, we show that a "typical" contraction $T$ contains the unit circle times the identity operator in the strong limit set of its powers, while $T^{n_j}$ converges weakly to zero along a sequence $\{n_j\}$ with density one. The continuous analogue is presented for isometric ang unitary $C_0$-(semi)groups.

自同态的部分交换半群中的算术增长

Arithmetic growth in partially commutative semigroups of endomorphisms

本文建立了簇的自同态的部分交换半群的度与高度增长率函数的渐近性，通过证明一般性结果并推广乒乓球引理，构造了新的自同态族。

圆周上有限生成非间隙作用的通用零熵优化

Generic Zero-Entropy Optimization for Finitely Generated Nonlacunary Actions on the Circle

针对圆周上有限生成非间隙乘法半群作用，证明利普希茨空间中开稠密子集的不变极值测度熵为零，且存在稠密G_δ子集使该结论对所有此类半群同时成立，结合了周期网格扰动估计与Rudolph–Johnson定理。

马丁边界与弱 disorder 中非线性乘法随机热方程

Martin Boundary and Invariant Fields of Multiplicative SHE

研究弱 disorder 中非线性乘法随机热方程的不变随机场，证明在自然二阶条件 下，正不变场与底层空间的有界正调和函数一一对应，继承马丁边界结构，并展示确定性热流收敛时随机演化收敛于相应不变场。

代数作用II. 群oid刚性

Algebraic actions II. Groupoid rigidity

研究代数作用的局部变换群oid的刚性，证明在特定条件下，群oid的全局化作用可相互嵌入，尤其在代数数论中，群oid能保持初始作用的同构性，解决了一个关于Cartan对同构的问题。

圆上Blaschke乘积的自由半群作用的混合性

Mixing for Free Semigroup Actions of Blaschke Products on the Circle

本文对单位圆盘内带不动点的有限Blaschke乘积及其自由半群作用的混合速率完成分类，揭示混合速率的非刚性及刚性条件，明确双指数混合的尖锐指数值。