Reversible maps and composites of involutions in groups of piecewise linear homeomorphisms of the real line

区间交换变换群中的可逆映射与对合乘积

Reversible Maps and Products of Involutions in Groups of IETS

本研究刻画了区间交换变换群$G_n$中的可逆与强可逆映射，证明二者等价，推导了对合乘积的数量上界，还分类了相关群的自由作用并给出实例。

四元数特殊线性群中两个对合的乘积

Product of two involutions in quaternionic special linear group

本文分类了四元数特殊线性群 $\mathrm{SL}(n,\mathbb{H})$ 和四元数射影线性群 $\mathrm{PSL}(n,\mathbb{H})$ 中的可逆与强可逆元素，并证明了元素可逆当且仅当它是两个斜对合或对合的乘积。

Factoring Formal Maps into Reversible or Involutive Factors

An element $g$ of a group is called reversible if it is conjugate in the group to its inverse. An element is an involution if it is equal to its inverse. This paper is about factoring elements as products of reversibles in the group $\mathfrak{G}_n$ of formal maps of $(\mathbb{C}^n,0)$, i.e. formally-invertible $n$-tuples of formal power series in $n$ variables, with complex coefficients. The case $n=1$ was already understood. Each product $F$ of reversibles has linear part $L(F)$ of determinant $\pm1$. The main results are that for $n\ge2$ each map $F$ with det$(L(F))=\pm1$ is the product of …

$\mathrm{SL}(n,\mathbb{C})$ 中的强可逆类

Strongly reversible classes in $\mathrm{SL}(n,\mathbb{C})$

本文研究了复特殊线性群$\mathrm{SL}(n,\mathbb{C})$中的强可逆元素，给出了元素可表示为两个对合乘积的充要条件，并据此完成了对强可逆共轭类的分类。

Conjugacy of reversible cellular automata and one-head machines

We show that conjugacy of reversible cellular automata is undecidable, whether the conjugacy is to be performed by another reversible cellular automaton or by a general homeomorphism. This gives rise to a new family of finitely-generated groups with undecidable conjugacy problems, whose descriptions arguably do not involve any type of computation. For many automorphism groups of subshifts, as well as the group of asynchronous transducers and the homeomorphism group of the Cantor set, our result implies the existence of two elements such that every finitely-generated subgroup containing both ha…

The group of reversible Turing machines: subgroups, generators and computability

We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only modify the tape at a bounded distance around the head, change the state and move the head in a bounded way. We study three natural subgroups arising in this model: the group of finite-state automata, which generalizes the topological full groups studied in topological dynamics and the theory of orbit-equivalence; the group of oblivious Turing machines whose movem…