AI 中文总结
研究超椭圆型某些线性对合的拓扑弱混合问题，通过考虑特定广义置换\(\sigma_{s,r}\)，证明了由其劳齐类中广义置换定义的典型线性对合的自然符号编码在有至少一个简单字母且相关亏格足够大时是拓扑弱混合的。
AI 中文摘要
线性对合是在一对不相交区间上定义的单射分段等距映射，由广义置换和长度向量给出的组合数据定义。如同区间交换变换，人们猜想除了某些组合数据外，典型的线性对合在测度理论上是弱混合的。在此方向上，可先探索线性对合拓扑模型的拓扑弱混合问题。本文证明了由某些超椭圆型广义置换定义的典型线性对合的自然符号编码是拓扑弱混合的。考虑广义置换\[ \sigma_{s,r} = \begin{pmatrix} 0 & A & 1 & 2 & \cdots & s & A & s+1 & s+2 & \cdots & s+r \\ s+r & \cdots & s+2 & s+1 & B & s & \cdots & 2 & 1 & B & 0 \end{pmatrix} \]。证明了由属于\(\sigma_{s,r}\)的劳齐类中的广义置换定义的典型线性对合的自然符号编码是拓扑弱混合的，前提是它至少有一个简单字母且其相关亏格足够大。
英文摘要
A linear involution is an injective piecewise isometry defined on a pair of disjoint intervals. They are defined by a combinatorial data given by a generalized permutation and a length vector. As it was done for interval exchange transformations, it is conjectured that, except for some combinatorial data, a typical linear involution is measure-theoretically weakly mixing. In this direction, one may first explore the question of topological weak mixing for topological models of linear involutions. In this article we prove topological weak mixing for the natural symbolic codings of typical linear involutions defined by some generalized permutations of hyperelliptic type. We consider the generalized permutation \[ σ_{s,r} = \begin{pmatrix} 0 & A & 1 & 2 & \cdots & s & A & s+1 & s+2 & \cdots & s+r \\ s+r & \cdots & s+2 & s+1 & B & s & \cdots & 2 & 1 & B & 0 \end{pmatrix}. \] We prove that the natural symbolic coding of a typical linear involution defined by a generalized permutation in the Rauzy class of $σ_{s,r}$ is topologically weakly mixing, provided that it has at least a simple letter and its associated genus is sufficiently large.
Comments23 pages, 1 figure, 1 table
